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Multi-layer diffusion model of photovoltaic installations

Tomasz Weron

TL;DR

The paper develops a two-layer agent-based framework that couples diffusion of photovoltaic adoption with opinion dynamics using a $q$-voter mechanism and independence on both layers. It analyzes two interaction variants, AND and OR, and shows that diffusion can succeed across a range of parameters independent of initial conditions, while a mean-field approximation captures the qualitative behavior observed in Monte Carlo simulations. The findings highlight how independence and cross-layer influence govern phase-like transitions among unadopted, adopted, and disordered states, and reveal differences between AND and OR in the ease and nature of transitions. The work provides a practical modeling approach to study PV diffusion and its interaction with public opinion, with implications for grid stability and renewable-energy policy.

Abstract

Nowadays, harmful effects of climate change are becoming increasingly apparent. A vital issue that must be addressed is the generation of energy from non-renewable and often polluting sources. For this reason, the development of renewable energy sources is of great importance. Unfortunately, too rapid spread of renewables can disrupt stability of the power system and lead to energy blackouts. One should not simply support it, without ensuring sustainability and understanding of the diffusion process. In this research, we propose a new agent-based model of diffusion of photovoltaic panels. It is an extension of the q-voter model that utilizes a multi-layer network structure. The novelty is that both opinion dynamics and diffusion of innovation are studied simultaneously on a multidimensional structure. The model is analyzed using Monte Carlo simulations and the mean-field approximation. The impact of parameters and specifications on the basic properties of the model is discussed. Firstly, we show that for a certain range of parameters, innovation always succeeds, regardless of the initial conditions. Secondly, that the mean-field approximation gives qualitatively the same results as computer simulations, even though it does not utilize knowledge of the network structure.

Multi-layer diffusion model of photovoltaic installations

TL;DR

The paper develops a two-layer agent-based framework that couples diffusion of photovoltaic adoption with opinion dynamics using a -voter mechanism and independence on both layers. It analyzes two interaction variants, AND and OR, and shows that diffusion can succeed across a range of parameters independent of initial conditions, while a mean-field approximation captures the qualitative behavior observed in Monte Carlo simulations. The findings highlight how independence and cross-layer influence govern phase-like transitions among unadopted, adopted, and disordered states, and reveal differences between AND and OR in the ease and nature of transitions. The work provides a practical modeling approach to study PV diffusion and its interaction with public opinion, with implications for grid stability and renewable-energy policy.

Abstract

Nowadays, harmful effects of climate change are becoming increasingly apparent. A vital issue that must be addressed is the generation of energy from non-renewable and often polluting sources. For this reason, the development of renewable energy sources is of great importance. Unfortunately, too rapid spread of renewables can disrupt stability of the power system and lead to energy blackouts. One should not simply support it, without ensuring sustainability and understanding of the diffusion process. In this research, we propose a new agent-based model of diffusion of photovoltaic panels. It is an extension of the q-voter model that utilizes a multi-layer network structure. The novelty is that both opinion dynamics and diffusion of innovation are studied simultaneously on a multidimensional structure. The model is analyzed using Monte Carlo simulations and the mean-field approximation. The impact of parameters and specifications on the basic properties of the model is discussed. Firstly, we show that for a certain range of parameters, innovation always succeeds, regardless of the initial conditions. Secondly, that the mean-field approximation gives qualitatively the same results as computer simulations, even though it does not utilize knowledge of the network structure.
Paper Structure (9 sections, 3 theorems, 34 equations, 8 figures, 1 algorithm)

This paper contains 9 sections, 3 theorems, 34 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Methods
  3. Simulation Model
  4. Mean-Field Approximation
  5. Results
  6. Simulation Details
  7. Simulation Model
  8. Mean-Field Approximation
  9. Conclusions

Key Result

Theorem 1

Let $c_A$ denote the concentration of positive adoption states and the probability of choosing an agent with a positive adoption state, and $c_S$ -- the concentration of positive opinions and the probability of choosing an agent with a positive opinion. Under the assumptions that these events are in and in the OR variant by:

Figures (8)

  • Figure 1: Data on solar installations in Poland in years 2019--2023, according to ARE (Polish energy market agency). More detailed data is being collected from 2021 onward, most likely due to the growing interest in the prosumer market. Data freely available at www.are.waw.pl.
  • Figure 2: Graphical representation of the model. The network consists of 2 layers: Square Lattice (SL, left side), on which adoption states $A_i$ are visible and two-dimensional Watts-Strogatz (WS2D, right side) with opinions $S_i$. Adoption states $A_i$ are represented by outer circles (green -- $A_i=+1$, red -- $A_i=-1$), while opinions -- by inner circles. Grey areas correspond to adoption states or opinions unknown to the target agent (marked with a dark blue circle). Groups of influence (of size $q=4$, marked with light blue circles) are constructed independently on each layer. In the given example, such a choice would be sufficient to change target's opinion ($S_{target} \to +1$) in the OR variant, but not in the AND variant, as unanimity is only achieved in one of the two groups of influence.
  • Figure 3: 10 simulated time trajectories of $c_A$ (red) and $c_S$ (blue) for different values of $p$: the AND (top), the OR variant (bottom). Values of $a_1$: $a_1=0.04$ (left), $a_1=0.16$ (right). First layer -- SL(N,1), second layer -- WS2D(N,1,0.2), size $N=2500$ and $a_2=0.5a_1$ in all cases.
  • Figure 4: 10 simulated time trajectories of $c_A$ (red) and $c_S$ (blue) for different values of $p$: the AND (top), the OR variant (bottom). Values of $a_1$: $a_1=0.04$ (left), $a_1=0.16$ (right). First layer -- SL(N,1), second layer -- WS2D(N,1,0.2), size $N=2500$ and $a_2=0.25a_1$ in all cases.
  • Figure 5: 10 simulated time trajectories of $c_A$ and $c_S$ versus numerically obtained time trajectory from Eqs. (\ref{['eq:pv_A']})-(\ref{['eq:pv_Sor']}), for different values of $p$ and $a_1$. The AND (top 4) and the OR variant (bottom 4). First layer -- SL(N,1), second layer -- WS2D(N,1,0.2), size $N=2500$ and $a_2=0.5a_1$ in all cases.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof