# National energy system modeling for decarbonization pathways considering distribution grids

Corentin Jacquier, Rémy Rigo-Mariani, Vincent Debusschere, Jean-Nicolas

Louis, Silvana Mima

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Corentin Jacquier, Rémy Rigo-Mariani, Vincent Debusschere, Jean-Nicolas Louis, Silvana Mima. National energy system modeling for decarbonization pathways considering distribution grids. Energy Strategy Reviews, 2025, 58, pp.101688. 10.1016/j.esr.2025.101688. hal-04991395

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# National energy system modeling for decarbonization pathways considering distribution grids

Corentin Jacquier<sup>a, *</sup>, Rémy Rigo-Mariani<sup>a</sup>, Vincent Debusschere<sup>a</sup>, Jean-Nicolas Louis<sup>b</sup>, Silvana Mima<sup>c</sup>

## ARTICLE INFO

Handling editor: Mark Howells

### Keywords:

- Energy system modeling

- Distribution grid

- Model coupling

- Optimization

- POLES

- Backbone

## ABSTRACT

Long-term planning tools compute investment in energy technologies to reach techno-economic objectives. These tools are typically used to define decarbonization pathways with the corresponding capacities for several types of technologies. However, for most renewable-based resources that are expected to be connected in distribution grids, the typical temporal and geographical resolutions of long-term models are deemed insufficient. They cannot capture the need for dynamic flexibility and/or grid reinforcement in distribution and thus the impact on the final costs or carbon emissions. This paper assesses the integration of distribution grids in long-term energy system planning. A coupling is proposed between the long-term model POLES, which computes the technology capacities to install each year up to 2050, with the dispatch/investment model Backbone, which optimizes the operation and the repartition of investments at finer resolutions. The coupling being computationally demanding, several simplifications of Backbone are investigated to find a trade-off with accuracy. Results show that simplifying the investment options and distribution grid models can achieve a 90 % precision on investment and energy generation compared to baseline optimization, with a computational time divided by 100. Finally, comparative simulations are run with the models coupling up to 2050 for the French power system with and without considering medium voltage distribution grids in addition to the transmission one. In the latter case, minor changes occur in the capacity investments. However, operational constraints increase the power system decarbonization cost by 20 % due to distribution constraints, underlining the necessity of considering finer temporal and geographical resolutions.

## 1. Introduction

In response to the need to curb global warming, the European Union (EU) pledged to achieve carbon neutrality by 2050. To this end, the EU envisioned potential decarbonization pathways [1], which consist of defining investment plans to limit greenhouse gas emissions while accounting for changes in energy usage (e.g., electrification). These studies conventionally rely on the use of long-term system models, with macroeconomic considerations based on partial equilibriums of different commodities [2]. The long-term models are limited by their coarse temporal and spatial granularities – e.g., yearly energy balances, at a national level. This endangers the relevance of the results obtained from the perspective of the electricity system. Accounting for daily/hourly variations is mandatory when assessing the need for storage

technologies and/or flexibilities, especially in the presence of variable renewable energy (VRE) (i.e., intermittent generation) [3], which affects the final operation or investment costs. These costs also depend on the location of the installed VRE in the electricity grids, which may require reinforcements in the form of increased lines or the capacity of the transformers. Thus, long-term models can be coupled with short-term representations of the system for better accuracy. In the power system community, these latter models usually take the form of optimization problems with finer temporal/geographical resolutions. The conventional objective is to minimize the system cost (operation/investment) while accounting for technical constraints along the simulation horizon at an hourly/subhourly resolution, with a nodal representation of the grid. The typical outputs are grid reinforcement, unit dispatch, and investment, which study needs to be simultaneous as they interact

<footnote><sup>a</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, G2Elab, 38000, Grenoble, France</footnote>

<footnote><sup>b</sup> VTT Technical Research Centre of Finland Ltd, Smart Energy and Built Environment, PO Box 1100, FI-90571, Oulu, Finland</footnote>

<footnote><sup>c</sup> GAEL, CNRS, Grenoble INP, INRA, Univ. Grenoble Alpes, 38000, Grenoble, France</footnote>

<footnote>* Corresponding author. Univ. Grenoble Alpes, CNRS, Grenoble INP, G2Elab, 38000, Grenoble, France.</footnote>

<footnote>E-mail address: corentin.jacquier@g2elab.grenoble-inp.fr (C. Jacquier).</footnote>
together (e.g., flexibility management in competition with grid rein-
forcement [4]). In long-term studies, the electrical grid is oftentimes
considered at a transmission level to represent national and/or
cross-border power flows (e.g. Ref. [5]). However, most VRE are ex-
pected to be connected at the distribution level, closer to the delivery
points [6]. Thus, it is necessary to increase the spatial resolution of the
short-term models to further enhance the relevance of decarbonization
pathways and identify the infrastructure bottlenecks. Yet, modeling only
the distribution grids is not sufficient, except for standalone distribution
grids (e.g., isolated areas [7]). For a complex grid like the French power
grid, the transmission grid highly influences the distribution grid
behavior and needs to be modeled. However, coupling long-term,
transmission grid and distribution grid models is complex and induces
long computation times. Developing techniques to integrate both
transmission and distribution grids is an active research subject [8,9],
but these algorithms are more suited to optimal power flows study on
small systems or short temporal periods [10].

To control computation time, a common solution in literature is to solve long-term investment and short-term operation problems successively. This approach was implemented for modeling the Medium Voltage (MV) distribution and High Voltage (HV) transmission grids to gain better insights into grid constraints and reinforcements [11]. Grid studies also go together with infrastructure changes at different voltage levels and techno-economic compromises between batteries, hydrogen storage, and grid reinforcement [12]. Böttcher et al. emphasize the need to consider the energy exchanges between transmission and distribution grids because of the additional costs they could incur regarding infrastructure reinforcement [13]. On a larger scale, finer granularity models of transmission grids in some European regions are used, however with suboptimal distribution grid management [14]. As these studies plan separately capacity investment and operation, they lack realism and tend to overestimate investment on distribution grids. Indeed, these grids must comply with the proposed capacity investments, whereas the grids can be too weak to host these capacities.

To overcome this limitation, investments and operation problems
must be considered jointly. This approach is explored by Allard et al. to
represent distribution grids in long-term studies, consisting of modeling
a few distribution grids (only three) to limit the computational time
[15]. However, not every transmission grid node is connected to a dis-
tribution grid. Therefore, it can create an imbalance between trans-
mission nodes, impoverishing the genericity of the results. This problem
is solved in Ref. [16], where Low Voltage (LV) distribution grids are
modeled alongside transmission grid in Germany. These distribution
grids cover the whole country, with a distinction between rural and
urban grids in each modeled region. However, contrary to Ref. [15],
investments are made in a single shot without considering different
maturation rates and constraints along the pathway to 2050, which
could lead to less realistic solutions [17].

The study presented in this manuscript extends the existing research
in distribution grids consideration in long-term planning, by imple-
menting simplifications in the optimization problem formulation. Seven
problem formulations are compared to find the best trade-off between
computation time and precision. These simplifications of the investment
options and distribution grid models enable the modeling of the MV and
HV grids in decarbonization pathways for France, up to 2050. With this
modeling, this study aims to assess the impact of adding MV distribution
grids in decarbonization pathways.

The main contributions of this paper are.

* A methodology that couples a long-term macroeconomic model and a short-term dispatch/investment tool to define decarbonization pathways for large power systems considering distribution grids.

* The investigation of different formulations for the dispatch/invest-
ment problem and their comparison with a baseline to find a trade-
off between computational time and precision.

* An assessment of the technical and economic impacts of the distri-
  bution grid modeling in decarbonization pathways up to 2050 for the
  French power system.

The paper is organized as follows: Section 2 focuses on the models for long-term and short-term coupling. Then, Section 3 presents the main equations of the optimization problem and the implemented simplified formulations. Finally, the results of the simplified problems and the long-term planning are discussed in Section 4.

## 2. Long-term and short-term models coupling

### 2.1. *Considered models*

Existing tools are used and modified to consider distribution grids in energy outlooks. Long-term models aim to assess the structural changes needed to reach a certain goal of the energy system (e.g., carbon neutrality) through scenarios where different sets of hypotheses are tested. Their broad vision of the energy sector allows them to evaluate the best energy vector for each usage. These models can also be classified as equilibrium models because they use economic equilibrium to estimate future prices and demands. The investment and dispatch methods can however differ from models, with a distinction between optimization and simulation problems. There exists a large variety of long-term models [3], but only four models (GEM-E3 [18], TIMES [19], POLES [20], and PRIMES [21]) were identified to enable a national representation of individual European countries. Among these four models, only two (GEM-E3 and POLES) consider a worldwide system, which is essential in an interconnected world. Finally, POLES appears more suited than GEM-E3 for modeling distribution grids in long-term energy studies through its higher temporal and spatial resolutions.

Similarly, short-term models are plethoric but do not perform
equally. These short-term models are optimization frameworks, which
can also solve long-term investment problems but lack the economics of
equilibrium models. While all can solve linear programming optimiza-
tion problems, a handful can handle mixed integer linear programming
to refine some aspects such as storage. Among the highly flexible models
(Backbone [22], Calliope [23], PyPsa [24], and SpineOpt [25]), only
Backbone and SpineOpt were identified to perform linearized optimal
AC power flow while being versatile enough ([26,27]) compared to tools
more adapted to grid modeling (e.g. Ref. [24]). While SpineOpt is
written in Julia, Backbone is implemented in GAMS and appears faster
[28]. As such, the present investigations rely on coupling the long-term
model POLES and the short-term dispatch model Backbone, as these
models appear the most suited for this work.

POLES is a global partial equilibrium model providing decarbon-
ization pathways [20]. It is annually enhanced by the European Com-
mission for its global energy and climate outlook [29], providing
insights into the current policies' efficiency concerning climate agree-
ments. POLES represents the world energy system divided into 66 re-
gions, with higher precision in Europe, where each EU country member
is represented at a national granularity. Within these 66 regions, energy
needs are computed from exogenous assumptions such as the GDP or
demography. To calculate the supply of each energy vector (electricity,
fossil fuels, ...), the simulation assesses the investment needs and the
energy generation through nodal energy balances. The generators are
selected based on their investment and operating costs, which are
internally computed. Energy generation comes from various sources:
fossil fuels, biomass, nuclear, hydro, solar, wind, or storage, with several
types of each (e.g., with and without carbon capture), for 30 technolo-
gies in total. The energy mix then impacts the environment with
greenhouse gases, pollutant emissions and the land use measurement.
Then, calculations along successive years (up to 2100) are performed
while updating knowledge on already installed assets and energy prices.

The Backbone model is an open-source optimization framework with
numerous features thoroughly detailed in Ref. [22]. Backbone can solve
operation and/or investment problems and can integrate various features depending on the needs. This model is reliable as it has been validated with a comparison with a commercial model [22]. This research uses Backbone capability to optimize the power plant installed capacities and their operation, along with the investment for distribution and transmission grid reinforcement and storage and VRE investments. The main equations of the model are presented in the next section. Those will be further simplified in the paper to propose trade-offs between computation time and precision of the results.

## 2.2. Coupling

The coupling of both long-term and short-term models relies on runs of POLES and Backbone over successive years up to 2050 (Fig. 1). At every iteration, POLES computes the energy demand for each sector (e.g., industry, residential). For each sector, POLES endogenously estimates the share supplied by each energy vector (e.g., electricity, gas) to fit climate objectives. Then, POLES computes the capacities to install at a national level for electricity generation technologies to supply the power demand. These capacities alongside the power demand, the operational and investment costs are finally sent to Backbone.

Backbone optimization of the power system is then performed along the same representative horizon as POLES (i.e., six representative days at an hourly resolution for a year [30]). However, the short-term modeling allows for a nodal representation of the electricity system to account for inter-region exchanges. In other words, Backbone optimally distributes and operates the VRE, and storage national capacities computed by POLES at a nodal resolution while accounting for potential grid reinforcements. The nodal capacities for non-VRE generation technologies are not defined optimally but with repartition keys from Ref. [5]. These repartition keys result from the current regional dispatch of thermal and hydropower plants, implicitly assuming their repartition on the territory will not change significantly. In the base implementation, only the transmission grid is considered in Backbone. This research aims to improve the relevance of the prospective studies with the introduction of the granularity of distribution grids. In the subsequent sections, distribution grids refer to MV grids, as LV grids are not being considered in this study, as their consideration would induce intractable computation times.

Specific attention is attached to the disaggregation step in the

methodology. This disaggregation relates to the distribution of both the load curve and the existing capacities (in 2023) from the national level provided by POLES down to the distribution nodes. Concerning the installed photovoltaic (PV) and wind turbine capacities in 2023, data for the chosen grids is not available. However, they can be estimated with the data of VRE installed in each town [31], sorted by region [32]. Furthermore, every node (transmission and distribution) inside a region is subject to the same weather, neglecting shading issues. Regarding load disaggregation, repartition keys built with data from Refs. [33,34] are used to distribute the load from each sector (industry, residential, etc.) in each region and voltage level (HV or MV). These repartition keys are based on the current regional distribution of sectorial loads and the population dynamics. Inside a distribution grid, the load is dispatched with another type of repartition key built with the grid dataset [35]. This dataset provides the annual power consumption for each MV node, so the regional load is dispatched to match these values. More details for the load disaggregation are provided in Appendix C. However, the hourly load variations are similar at each node in the same MV grid. Therefore, with a low diversity factor, peak demand for each node happens at the same timestep, which is more constraining for the grid.

## 3. Short-term model simplification

The objective of the short-term model is to optimize the installed capacities of generation units and their operation to minimize the overall yearly cost. The operation of the units ($u \in U$) of 30 technologies (subsets for VRE: $U^{vre}$, storage: $U^{sto}$, and thermal: $U^{th}$ with $U = U^{th} \cup U^{sto} \cup U^{vre}$) takes the form of an optimization problem along the simulation period – representative days (set $t \in T$). The considered HV (transmission) and MV (distribution) grids are represented through a set of nodes ($n \in N = N^{hv} \cup N^{mv}$), either in distribution grids ($n^{mv} \in N^{mv}$) and or in the transmission grid ($n^{hv} \in N^{hv}$), and connecting lines/transformers ($l \in L$).

This work focuses on the French grid for the illustration, with 14 transmission nodes, but the presented methodology applies to other countries. Most of the work carried out in this paper consists of refining the nodal resolution in Backbone while accounting for distribution grids, as shown in Fig. 2.

Each transmission grid node is connected to one MV grid. This distribution grid accounts for the hundreds of existing ones in the considered region, so a scaling factor is set at the interface between the distribution and the transmission grids to enable the use of a realistic MV grid topology [35]. The fourteen regions are populated with only one type of distribution grid: an urban grid in the Paris region, and a rural grid elsewhere, as it represents the grid type with the most consumption in the respective regions (in the Paris region, 90 % of the load is urban whereas in the rest of France it is around 25 % [31]). The chosen rural and urban grids are respectively composed of 37 and 24 nodes (505 distribution grid nodes in total), are both radial and the grids' peak consumption is around 2 MW. The chosen grids emerge from the dataset [35] where 300 representative MV grids are defined. The characteristics (aerial share, resistance, length, etc.) of these grids regarding their rurality have been studied to provide grids representative of their category (rural, urban). Increasing the number of nodes per MV grid, or the number of MV grids per transmission node leads to intractable computation time.

### 3.1. Baseline formulation

#### 3.1.1. Objective

The objective of the short-term model Backbone is to optimize the installed capacities ($p_{n.u}^{cap}$) of VRE and storage units along with the operating power of every unit - ($p_{n.u,t}^{gen}$) being the power generated by unit $u$ connected at node $n$ at time $t$. The objective function (1) considers the generation cost of every technology ($\pi_u^{gen}$, null for VRE and storage)

<figure><img src="image_3.png" /><figcaption>Fig. 1. POLES and Backbone yearly coupling methodology.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Modeling of HV and MV grid in France.</figcaption></figure>

while accounting for an annuity factor ($\alpha_u^{cap}$) and an installation cost ($\pi_{n,u}^{cap}$). The grid reinforcement ($f_l^{cap}$) is also considered with the possibility of upgrading line and/or transformer capacities, with a capacity cost ($\pi_l^{cap}$) and with an annuity factor ($\alpha_l^{cap}$). To improve transparency, the main cost values are displayed in Appendix B. The timestep duration of one hour is noted $\Delta t$. As the yearly generation is represented by six representative days, a weight ($\omega_l^{day}$) is used, the sum of weights for all days being equal to 365. The weights are issued from the clustering method used to define the representative days [30]. Note that variables are defined with a lowercase Latin letter, while parameters with an uppercase Latin letter or a Greek letter. A nomenclature is provided in Appendix A.

$$ 
\begin{aligned}
obj = & \sum_{n \in N} \sum_{u \in U^{th}} \sum_{t \in T} P_{n,u,t}^{gen} \times \pi_u^{gen} \times \omega_t^{day} \times \Delta t + \\
& \underbrace{\sum_{n \in N} \sum_{u \in U^{re}, U^{sto}} P_{n,u}^{cap} \times \pi_u^{cap} \times \alpha_u^{cap}}_{\text{operating cost of thermal generation}} \\
& \underbrace{\sum_{l \in L} f_l^{cap} \times \pi_l^{cap} \times \alpha_l^{cap}}_{\text{cost of grid reinforcement}}
\end{aligned}
\quad (1) $$

### 3.1.2. Generation units

The output power of thermal generation units (e.g., coal, oil, gas, nuclear, biomass) shall be lower than the upper bound defined by the existing rated power ($P_{n,u}^{cap0}$). Maintenance of aging plants, not always producing at their installed capacity, is modeled with a daily maintenance rate ($M_{n,u,t}$), limiting the output power of units (2). Thermal units are also dynamically constrained by a maximal ramp rate ($R_{n,u}^{rampMax}$), limiting their output variation (3).

$$ 0 \le P_{n,u,t}^{gen} \le P_{n,u}^{cap0} \times (1 - M_{n,u,t}) \quad \forall n \in N, u \in U^{th}, t \in T \quad (2) $$

$$ -\Delta t \times R_{n,u}^{rampMax} \times P_{n,u}^{cap0} \le p_{n,u,t}^{gen} - p_{n,u,t-1}^{gen} \le \Delta t \times R_{n,u}^{rampMax} \times P_{n,u}^{cap0} \quad (3) $$
$$ \forall n \in N, u \in U^{th}, t \in T $$

For the renewable-based units, the output power follows normalized generation profiles depending on the weather ($W_{n,u,t}$) along the simulation horizon (4). Investments consist of increasing the existing

capacity $P_{n,u}^{cap0}$ with a continuous variable $p_{n,u}^{cap}$. Also, the ability to curtail the generation ($p_{n,u,t}^{curt}$) is considered if needed.

$$ P_{n,u,t}^{gen} + P_{n,u,t}^{curt} = (P_{n,u}^{cap0} + p_{n,u}^{cap}) \times W_{n,u,t} \quad \forall n \in N, u \in U^{vre}, t \in T \quad (4) $$

### 3.1.3. Storage units

Storage units gather hydro storage and batteries, the latter being installable at every node (on both HV and MV grids), if necessary. The charging ($p_{n,u,t}^{ch}$) and discharging ($p_{n,u,t}^{disch}$) powers are limited by the (existing or newly) installed capacity (5). The state of charge ($e_{n,u,t}$) is updated (6) following a standard linear model that accounts for the round-trip efficiency ($\eta_{n,u}$) and shall remain within specified limits (7). The storage capacity investment is linked to the storage power investment to increase the existing capacity ($E_{n,u}^{cap0}$) as storage units are assumed to have a standard power-to-energy ratio of 0.5. It can be noted that only intra-day storage is modeled, inter-day and seasonal storage being more computationally demanding [36].

$$ 0 \le p_{n,u,t}^{ch} \times p_{n,u,t}^{disch} \le P_{n,u}^{cap0} + p_{n,u}^{cap} \quad \forall n \in N, u \in U^{sto}, t \in T \quad (5) $$

$$ e_{n,u,t} - e_{n,u,t-1} = \Delta t \times (p_{n,u,t}^{ch} \times \eta_{n,u} - P_{n,u,t}^{disch}) \quad \forall n \in N, u \in U^{sto}, t \in T \quad (6) $$

$$ 0 \le e_{n,u,t} \le E_{n,u}^{cap0} + p_{n,u}^{cap} \times 2 \times 1h \quad \forall n \in N, u \in U^{sto}, t \in T \quad (7) $$

### 3.1.4. Transmission grid modeling

Within the nodal representation in Backbone, a difference is made between the modeling of the transmission and distribution grids. The transmission grid model relies on a DC power flow, with the active power flowing in lines ($f_{l,t}$) determined by the nodal balance in (8), with line losses ($\eta_l$). The nodal load ($P_{n,t}^{load}$) comes from the disaggregation of the national load curve considered in POLES. The power flow within a line shall not exceed its rated capacity ($E_{l}^{cap0}$) which can be increased with investment (9). Transformers between transmission and distribution nodes are represented as lines. That nodal balance accounts for the subsets of incoming lines at every node ($in(n)$) and outgoing links (out(n)).

$$ \underbrace{\sum_{l \in U^{in}(n) \subset L} (f_{l,t} \times \eta_l)}_{\text{import/export balance}} - \underbrace{\sum_{l \in U^{out}(n) \subset L} f_{l,t}}_{\text{nodal load}} = \underbrace{P_{n,t}^{load} + \sum_{u \in U^{sto}} p_{n,u,t}^{ch}}_{\text{nodal generation}} - \underbrace{\sum_{u \in U^{th}, U^{vre}} P_{n,u}^{gen} - \sum_{u \in U^{sto}} P_{n,u}^{disch}}_{\text{nodal generation}} \quad \forall n \in N, t \in T \quad (8) $$
$$-F_{l}^{\text{cap0}} - f_{l}^{\text{cap}} \le f_{l,t} \le F_{l}^{\text{cap0}} + f_{l}^{\text{cap}} \quad \forall l \in L, t \in T \quad (9)$$

### 3.1.5. Distribution grid modeling

The baseline model accounts for voltage drops and reactive power flows in distribution grids. It follows a standard linear DistFlow model [37] tailored for radial grids (10). The variable ($v_{n,t}$) denotes the square value of the node voltage. In addition to the active power, reactive power flow ($f_{l,t}^Q$) appears in this AC load flow alongside the line resistance ($R_l$) and reactance ($X_l$). A reactive power balance is also introduced (11), to feed the reactive load ($P_{n,t}^{load-Q}$) assumed proportional to the active one (power factor of 0.92 [35]), with reactive power generation ($P_{n,t}^{gen-Q}$) coming from distributed assets (inverter-based generation). The introduction of reactive power changes line power limits, for which a linearized model of the squared real limits is used, as in Ref. [38].

$$v_{n^{mv},t} = 2 \times R_l \times f_{l,t} + 2 \times X_l \times f_{l,t}^Q + v_{n^{mv},t} \quad \forall t \in T, \\ n^{mv} \in N^{mv}, m^{mv} \in N^{mv}, l \in \text{out}(n^{mv}) \cap \text{in}(m^{mv}) \subset L \quad (10)$$

$$\sum_{l \in \text{out}(n^{mv}) \subset L} f_{l,t}^Q - \sum_{l \in \text{out}(n^{mv}) \subset L} f_{l,t}^Q = P_{n^{mv},t}^{\text{loadQ}} - \sum_{u \in U^{vre}, u \in U^{sto}} P_{n^{mv},u,t}^{\text{genQ}} \\ \forall n^{mv} \in N^{mv}, t \in T \quad (11)$$

### 3.1.6. Dispatching investments between distribution and transmission grids

The choice between transmission or distribution grids for the installation of VRE is not straightforward, as more power plants in distribution grids mean that production is closer to consumption, but also that reverse flows in transformers are higher [39,40]. Therefore, with different advantages for capacity installation in distribution and transmission nodes, the optimization process selects the best trade-off from the system point of view. Thus, Backbone optimizes the repartition of new VRE and storage plants (12), which national capacity is provided by POLES ($P_u^{POLES}$). Backbone minimizes operation and investment costs in both distribution and transmission grids. However, the VRE installation in the transmission grid is constrained. Indeed, the French energy outlooks of both transmission [41] and distribution grids [42] lead to the conclusion that more than two-thirds of the VRE should be installed in distribution nodes. It sets a higher limit on the new capacity of VRE installation in the transmission grid (13).

$$\sum_{n \in N} p_{n, u}^{\text{cap}} = P_{u}^{\text{POLES}} \quad \forall u \in U^{vre} \cup U^{sto} \quad (12)$$

$$\sum_{n^{hv} \in N^{hv}} p_{n^{hv}, u}^{\text{cap}} \le \frac{1}{3} P_{u}^{\text{POLES}} \quad \forall u \in U^{vre} \quad (13)$$

The arbitrage between investment in transmission or distribution nodes also accounts for discrepancies in investment cost ($\pi_{n,t}^{cap}$) for some technologies. For instance, battery projects are 12 % [43] and VRE projects are 30 % more expensive in distribution grids [44].

### 3.2. Simplifications assumptions

The Backbone optimization is run using GAMS with the CPLEX solver, on an i5-1145G7 processor with 16 GB RAM. With the baseline formulation, a single run (i.e., one year represented by six days), the computational time is about 8 h, so a simulation targeting 2050 would last more than nine days. Additional assumptions/simplifications to the problem formulation are thus needed to reach a satisfying compromise between the accuracy of the returned solution and calculation time. It can be noted that the baseline formulation dispatch has been validated after comparison with 2023 data from Ref. [34].

#### 3.2.1. DC modeling

The first simplification involves using a DC load flow for the distribution grids, like the transmission grid. With only the active power considered, the number of corresponding optimization variables is divided by three compared to the DistFlow originally used to represent voltage and reactive flow in the distribution grids.

#### 3.2.2. Storage aggregation

Each storage unit requires three variables (charge, discharge, and state of charge) defined along the optimization horizon $T$, significantly impacting computational time. In the baseline formulation, there can be as many storage units as the total number of nodes. To further simplify that formulation, a limited number of nodes can be selected in every distribution grid to potentially host a storage unit. Thus, in every distribution grid, nodes are ranked in descending order with their peak electrical load. Tests are performed while selecting only the first 1, 5, and 10 nodes to install storage units.

#### 3.2.3. Investment Allocation

In the baseline formulation, the number of investment variables is proportional to the number of nodes. An alternative simplification is envisioned with a unique capacity investment variable in each transmission node for each technology subject to investment (VRE, storage). This investment variable provides in each region (i.e., transmission grid node) the share ($s_{n^{hv},u}$) of the regional capacity installed in MV grids (14), the remaining share being for capacity investment in the transmission grid (15). Then, inside a distribution grid, a repartition key ($\beta_{n,u}$) is used to distribute the chosen share among the nodes. Thus, investment variables are reduced to the number of transmission nodes (14 instead of 505).

$$P_{n^{mv},u}^{\text{cap}} = s_{n^{hv},u} \times \beta_{n^{mv},u} \times P_{u}^{\text{POLES}} \quad \forall n^{hv} \in N^{hv}, n^{mv} \in N^{mv}, \\ u \in U^{vre} \cup U^{sto} \quad (14)$$

$$\forall n^{hv}, u : P_{n^{hv},u}^{\text{cap}} = (1 - s_{n^{hv},u}) \times P_{u}^{\text{POLES}} \quad \forall n^{hv} \in N^{hv}, u \in U^{vre} \cup U^{sto} \quad (15)$$

## 4. Results

### 4.1. Computational time reduction

First, different simplifications of the baseline Backbone formulation are tested with the optimization problem run over a single year. The simulation accounts for the national load profiles and maximum installed capacities from POLES outputs in 2050. As displayed in Table 1, the formulations reduce differently the number of variables, per type. Every problem simplification is assessed regarding its precision compared to the baseline formulation.

Three metrics are introduced with the generated/curtailed energy, installed capacity, and the localization of investments (nodes of distribution grids) to measure the average deviation with the baseline. The results are displayed in Fig. 3, highlighting the necessary trade-off between accuracy and computational time.

Overall, the energy/curtailment precision remains high (over 80 %) for the investigated simplifications. The capacity investment precision, measuring the investment subject to optimization (grid reinforcement, share of assets in each voltage level) is close to 100 %. This is expected as the generated energy and installed capacities are driven by the load to supply, which is unchanged over all the simulations. However, the localization of these investments differs more significantly (precision down to 70 %) when simplifying the investment problem. More importantly, the computational time varies in a magnitude of 100, with only 2 min for the most simplified version of the investment problem
<table><caption>**Table 1**
Problem formulations.</caption><thead><tr><th colspan="4">Problem formulation</th><th rowspan="2">Nb. of grid variables</th><th rowspan="2">Nb. of operation variables</th><th rowspan="2">Nb. of capacities variables</th></tr><tr><th>Name</th><th>Distribution grid model</th><th>Storage aggregation</th><th>Investment Allocation</th></tr></thead><tbody><tr><td>Baseline</td><td>AC</td><td>-</td><td>-</td><td>111,904</td><td>442,950</td><td>1571</td></tr><tr><td>F1</td><td>AC</td><td>✓ (10 nodes)</td><td>-</td><td>111,904</td><td>274,825</td><td>1206</td></tr><tr><td>F2</td><td>DC</td><td>-</td><td>-</td><td>37,960</td><td>307,446</td><td>1571</td></tr><tr><td>F3</td><td>DC</td><td>✓ (10 nodes)</td><td>-</td><td>37,960</td><td>175,681</td><td>1206</td></tr><tr><td>F4</td><td>DC</td><td>✓ (5 nodes)</td><td>-</td><td>37,960</td><td>150,411</td><td>1136</td></tr><tr><td>F5</td><td>DC</td><td>✓ (1 node)</td><td>-</td><td>37,960</td><td>130,195</td><td>1080</td></tr><tr><td>F6</td><td>DC</td><td>-</td><td>✓</td><td>37,960</td><td>280,801</td><td>42</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Precision and computation time for the investigated formulations.</figcaption></figure>

(F5, i.e., with DC power flow and one node storage aggregation).

## 4.2. Impact of considering distribution grids on prospective scenarios results

From the performed tests, the fastest implementation (F5 in Table 1) of the Backbone investment/operation is coupled with POLES to perform long-term simulations with successive yearly runs, as this implementation provides sufficient precision on the main metrics studied in the subsequent results (capacity, prices, energy). POLES provides the power demand for each year to limit the global temperature increase to 1.5 °C at the end of the century, which is quite challenging for the energy system. The decarbonization comes with a significant increase in power demand, especially in the transportation sector as displayed in Fig. 4. For the other sectors, the electrification impact is more contained as in parallel there is an increase of the system efficiency (e.g., increasing thermal insulation in buildings).

In Fig. 5, the integration of distribution grids in the simulation model in addition to transmission one (denoted as T&D) is compared with an initial simulation in which only the transmission grid is considered (denoted as T) – i.e. all the installed capacities are connected to the high voltage nodes without any distribution grid consideration.

The obtained results for both T and T&D display similar values for the generated energy and installed capacity during the simulated years. This is due to the supplied load, which is the same in both approaches (reaching up to 700 TWh in 2050 as proposed by POLES). However,

when distribution grids are considered, more curtailment of VRE occurs due to the impossibility, without further costly grid reinforcements, of consuming the generated energy at some point in time. It can be noted that fossil capacity keeps growing whereas it remains unused. The phenomenon can partially be explained by the lack of reserve modeling [45], to reduce the computation time. Another limitation explaining the lack of fossil-based electricity production is the modeling of only six representative days, whereas it is advised to model at least twenty representative days to properly capture a year [46].

A focus on the invested capacities (Fig. 6) shows significant grid upgrades, especially in the 2030s to face the increasing power demand due to electrification. This result emphasizes that decarbonization relies not only on VRE but also significantly on distribution grid reinforcements. Fig. 6 also shows the emergence of flexibility investments after 2035 to manage the growing VRE capacity. Batteries emerge after 2045, but distributed batteries remain marginal because of their higher cost. An interesting point is the share of VRE investments in distribution grids being constant over successive years. This is due to the assumptions from Section 3, limiting the installation of VRE in transmission grids to 1/3 of the total investments. Therefore, investments in VRE connected to the transmission grid always hit the limit, meaning they are more interesting than investments in distribution grids.

If the differences in power and energy remain negligible between T and T&D models, considering the distribution grids impacts significantly costs, as shown in Table 2. With more capacities in distribution grids and line/transformer reinforcement, the overall system cost indeed strongly increases (+20 %). This cost difference comes from the operation (fuel cost) because as reinforcing distribution grid is expensive, VRE curtailment is higher there, thus controllable plants with higher operational costs are used more (c.f. Fig. 5). The investment costs also present a difference because, for the same capacity, transmission grid plants benefit from scale effect and are then less expensive. Considering distribution grids in prospective scenarios provides then more realistic decarbonization costs.

Further analyses are needed to strengthen the results on grid investments because the main French distribution system operator (ENEDIS, handling 95 % of the French distribution grid) foresees that distribution grids will require more than 500 million euros in investment per year [42] whereas in this study only around one billion euros are needed for MV grids for the whole 2023–2050 period. The global cost is of the same order of magnitude as other references [41], but model calibration may improve cost repartitions. The exclusion of LV grids from this study can partially explain the lower cost compared to studies considering both LV and MV grids because, similarly to MV grids, LV grids will face major changes [42]. The absence of LV grids in this study is a major limitation as modeling these grids could widen even further the gap with conventional studies only considering the HV grid.

## 5. Conclusion

This paper proposes a methodology to consider distribution grids in long-term energy system planning studies of large power systems (countries, continents). This enables a better representation of the

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Sectorial power demand in France.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Installed capacity and annual power production evolution in France from 2025 to 2050 for transmission grid only (T) and transmission & distribution grid (T&D) models.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Detailed annual investments in VRE, grid, and flexibility.</figcaption></figure>

<table><caption>**Table 2**
Investment and operation costs from 2023 to 2050 for two scenarios.</caption><thead><tr><th></th><th>Power Plant Cost</th><th>Operating Cost</th><th>Grid Reinforcement</th><th>Total</th></tr></thead><tbody><tr><th>T</th><td>396 G€</td><td>257 G€</td><td>0 G€</td><td>653 G€</td></tr><tr><th>T&D</th><td>439 G€</td><td>357 G€</td><td>0.8 G€</td><td>796 G€</td></tr></tbody></table>

distribution grids' operational constraints which induces an adaptation of the decarbonization pathways. In the presented research, the distribution grid modeling in long-term studies is enabled by coupling a long-term partial equilibrium model, POLES, and Backbone, a versatile energy framework.

This coupling method induces a significant computational burden; therefore, several problem formulations are compared to find the best compromise between computation time and precision. The DC load flow with one battery aggregation performs well with a computation time divided by 100, in exchange for a precision reduction by a maximum of 10 %. This modeling enables the impact assessment of distribution grids' consideration in long-term studies. On a reference prospective scenario, the comparison between transmission grid only and transmission with distribution grid modeling results in minor changes in the capacity investment plan. However, higher VRE curtailments, when considering distribution grids, lead to a significant increase in the system cost (20 %). It questions the impact on the end-user electricity price and the funding of the energy transition. This high curtailment rate also raises questions about the economic viability of such systems, as without a clearer view on distribution grid impacts in long-term planning, VRE installation profitability could seriously be overestimated. Therefore, policies to encourage demand-side flexibility could reduce curtailment while reducing the cost of replacing this curtailed energy.

Despite these promising results, the 14 distribution grids modeled

represent a tiny part of the distribution grids variety. Further work is therefore needed to be able to model a representative set of grids while preserving an acceptable computation time. A sensitivity analysis on the distribution grid could also provide the key parameters to define the most suited grids for VRE integration. Finally, a European scale simulation could be valuable to assess the impact of the flexibility brought by power exchanges between countries on VRE installation in distribution grids.

### CRediT authorship contribution statement

**Corentin Jacquier:** Methodology, Software, Data curation, Visualization, Writing – original draft. **Rémy Rigo-Mariani:** Conceptualization, Methodology, Writing – review & editing. **Vincent Debusschere:** Conceptualization, Methodology, Writing – review & editing, Supervision. **Jean-Nicolas Louis:** Conceptualization, Methodology, Resources, Writing – review & editing. **Silvana Mima:** Conceptualization, Methodology, Resources, Writing – review & editing.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

Jean-Nicolas Louis acknowledges the Research Council of Finland for providing funding for this research with the Nexus4EU project (333076) and the 2IMatch project (358648).
## Appendix A. Nomenclature

<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>Sets</td><td></td></tr><tr><td>l &isin; L</td><td>Set of lines</td></tr><tr><td>n &isin; N</td><td>Set of nodes</td></tr><tr><td>t &isin; T</td><td>Set of timesteps</td></tr><tr><td>u &isin; U</td><td>Set of units</td></tr><tr><td>Subsets</td><td></td></tr><tr><td>in(n) &subset; L</td><td>Lines to node n</td></tr><tr><td>out(n) &subset; L</td><td>Lines from node n</td></tr><tr><td>N<sup>hv</sup> &subset; N</td><td>HV grid nodes</td></tr><tr><td>N<sup>mv</sup> &subset; N</td><td>MV grid nodes</td></tr><tr><td>U<sup>vre</sup> &subset; U</td><td>Renewable units</td></tr><tr><td>U<sup>stv</sup> &subset; U</td><td>Storage units</td></tr><tr><td>U<sup>th</sup> &subset; U</td><td>Thermal units</td></tr><tr><td>Parameters</td><td></td></tr><tr><td>E<sub>n,ui</sub><sup>cap0</sup></td><td>Storage capacity (MWh)</td></tr><tr><td>F<sub>l</sub><sup>cap0</sup></td><td>Line capacity (MVA)</td></tr><tr><td>M<sub>n,ui</sub></td><td>Maintenance rate</td></tr><tr><td>P<sub>n,ui</sub><sup>cap0</sup></td><td>Nodal installed capacity (MW)</td></tr><tr><td>p<sub>n,t</sub><sup>load</sup></td><td>Nodal power demand (MW)</td></tr><tr><td>p<sub>n,t</sub><sup>load_Q</sup></td><td>Reactive power demand (Mvar)</td></tr><tr><td>P<sub>u</sub><sup>POLES</sup></td><td>New capacity to install (MW)</td></tr><tr><td>R<sub>l</sub></td><td>Line resistance (&Omega;)</td></tr><tr><td>R<sub>u</sub><sup>rampMax</sup></td><td>Maximal ramp rate (h<sup>-1</sup>)</td></tr><tr><td>W<sub>n,ui,t</sub></td><td>Renewable weather profile (W/Wp)</td></tr><tr><td>X<sub>l</sub></td><td>Line reactance (&Omega;)</td></tr><tr><td>&alpha;<sub>l</sub><sup>cap</sup></td><td>Annuity factor for lines</td></tr><tr><td>&alpha;<sub>u</sub><sup>cap</sup></td><td>Annuity factor for units</td></tr><tr><td>&beta;<sub>n,ui</sub></td><td>Repartition key for MV investments</td></tr><tr><td>&Delta;t</td><td>Timestep duration (h)</td></tr><tr><td>&eta;<sub>n,ui</sub></td><td>Storage efficiency</td></tr><tr><td>&eta;<sub>l</sub></td><td>Line losses factor</td></tr><tr><td>&pi;<sub>n,ui</sub><sup>cap</sup></td><td>Unit investment cost (&euro;/MW)</td></tr><tr><td>&pi;<sub>l</sub><sup>cap</sup></td><td>Line investment cost (&euro;/MVA)</td></tr><tr><td>&pi;<sub>u</sub><sup>gen</sup></td><td>Operational cost (&euro;/MWh)</td></tr><tr><td>&omega;<sub>t</sub><sup>day</sup></td><td>Weight for representative days</td></tr><tr><td>Variables</td><td></td></tr><tr><td>e<sub>n,ui,t</sub></td><td>Energy stored (MWh)</td></tr><tr><td>f<sub>l,t</sub></td><td>Power flow (MW)</td></tr><tr><td>f<sub>l,t</sub><sup>Q</sup></td><td>Reactive power flow (Mvar)</td></tr><tr><td>f<sub>l</sub><sup>cap</sup></td><td>Line capacity investment (MVA)</td></tr><tr><td>P<sub>n,ui</sub><sup>cap</sup></td><td>Capacity investment (MW)</td></tr><tr><td>P<sub>n,ui,t</sub><sup>ch</sup></td><td>Battery charging power (MW)</td></tr><tr><td>P<sub>n,ui,t</sub><sup>disch</sup></td><td>Battery discharging power (MW)</td></tr><tr><td>P<sub>n,ui,t</sub><sup>curt</sup></td><td>Curtailed power (MW)</td></tr><tr><td>P<sub>n,ui,t</sub><sup>gen</sup></td><td>Power generation (MW)</td></tr><tr><td>P<sub>n,ui,t</sub><sup>gen_Q</sup></td><td>Reactive power generation (Mvar)</td></tr><tr><td>s<sub>n,ui</sub></td><td>Share of MV capacity installation</td></tr><tr><td>v<sub>n,t</sub></td><td>Squared voltage (V<sup>2</sup>)</td></tr></tbody></table>

## Appendix B. Price examples

<table><caption>Table 3
ost hypotheses</caption><thead><tr><th></th><th>2025</th><th>2030</th><th>2035</th><th>2040</th><th>2045</th><th>2050</th><th>Source</th></tr></thead><tbody><tr><td>PV (€/kW)</td><td>555</td><td>452</td><td>402</td><td>350</td><td>318</td><td>301</td><td>POLES</td></tr><tr><td>Wind turbine (€/kW)</td><td>1046</td><td>917</td><td>843</td><td>785</td><td>740</td><td>706</td><td>POLES</td></tr><tr><td>Battery (€/kW)</td><td>1046</td><td>889</td><td>641</td><td>513</td><td>443</td><td>396</td><td>POLES</td></tr><tr><td>Transmission grid line (€/MW/km)</td><td>824</td><td>-</td><td>-</td><td>-</td><td>-</td><td>824</td><td>[47]</td></tr><tr><td>Urban distribution grid line (€/MW/km)</td><td>4609</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4609</td><td>[48]</td></tr><tr><td>Rural distribution grid line (€/MW/km)</td><td>3920</td><td>-</td><td>-</td><td>-</td><td>-</td><td>3920</td><td>[49]</td></tr><tr><td>Transformer (€/kW)</td><td>16</td><td>-</td><td>-</td><td>-</td><td>-</td><td>16</td><td>[49]</td></tr></tbody></table>
## Appendix C. Load disaggregation methodology

The regional load ($P_{y,r,s,t}^{loadReg}$) in year $y \in Y$, in region $r \in R$, for sector $s \in S$ is defined with the national load $P_{y,s,t}^{loadNat}$, the current and initial population pop$_{y,r}$, and the repartition key for the regional distribution of the sectorial demand $\lambda_{r,s}^{sector}$.

$$P_{y,r,s,t}^{loadReg} = P_{y,s,t}^{loadNat} \times \frac{\lambda_{r,s}^{sector} \times pop_{y,r}}{pop_{2023,r} \times \sum_{r \in R} \lambda_{r,s}^{sector} \times pop_{2023,r}} \quad \forall y \in Y, r \in R, s \in S, t \in T$$

This repartition key ($\lambda_{r,s}^{sector}$) is built with the initial data that can be found in Ref. [31].

$$\lambda_{r,s}^{sector} = \frac{P_{y,r,s,t}^{loadReg}}{P_{y,r,s,t}^{loadNat}} \quad \forall r \in R, s \in S, t \in T$$

The regional load is then split between the HV and MV nodes inside a region ($r \in reg(n) \subset R$) with a repartition key for the share of each sector consumption in HV level ($\lambda_s^{HV}$, e.g. $\lambda_{r,s}^{HV}$ residential = 0 because the residential demand is on distribution grids).

$$P_{y,n^{hv},t}^{load} = \sum_{s \in S, r \in reg(n^{hv})} P_{y,r,s,t}^{loadReg} \times \lambda_s^{HV} \quad \forall y \in Y, n^{hv} \in N^{hv}, t \in T$$

The load in MV is more complex as each MV grid represents all the MV grids in its region. Therefore, the number of distribution grids in each region ($k_r^{distrib}$) is used.

$$P_{y,n^{mv},t}^{load} = \lambda_{n^{mv}}^{mvLoad} \times \sum_{s \in S, r \in reg(n^{mv})} P_{y,r,s,t}^{loadReg} \times \frac{(1 - \lambda_s^{HV})}{k_r^{distrib}} \quad \forall y \in Y, n^{mv} \in N^{mv}, t \in T$$

Finally, the repartition key for the share of a MV node load inside a MV grid load ($\lambda_{n^{mv}}^{mvLoad}$) is defined similarly to the regional load, but with data from Ref. [35].

$$\lambda_{n^{mv}}^{mvLoad} = \sum_{t \in T, r \in reg(n^{mv})} \frac{P_{2023,n^{mv},t}^{load} \times k_r^{distrib}}{\sum_{s \in S} P_{2023,r,s,t}^{loadReg} \times (1 - \lambda_s^{HV})} \quad \forall n^{mv} \in N^{mv}$$

## Data availability

Data will be made available on request.

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