pith. sign in

arxiv: 2605.02894 · v1 · submitted 2026-03-08 · 🧮 math.DS · cs.SY· eess.SY

Analysis of a Stochastic Energy Supply and Demand Model with Renewable Integration

S.O.Edeki , S.Noeiaghdam , L. Hairong , S. Kaennakham , I. D. Ezekiel This is my paper

Pith reviewed 2026-05-15 15:07 UTC · model grok-4.3

classification 🧮 math.DS cs.SYeess.SY
keywords stochastic differential equationsenergy supply-demand modelrenewable integrationalmost sure stabilitymoment boundednessglobal existenceEuler-Maruyama schemestochastic persistence
0
0 comments X

The pith

A stochastic Ito model for energy supply and demand with renewables admits globally positive, moment-bounded, and almost surely exponentially stable solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a deterministic energy supply-demand model to include multiplicative stochastic noise from market and climate uncertainties. It rigorously proves global existence and uniqueness of positive solutions, along with their moment boundedness and stochastic persistence. Stability analysis using matrix inequalities shows almost sure exponential stability under appropriate parameter conditions. These results indicate that stochastic effects change effective capacity but do not destroy the system's core bounded and stable behavior. This matters for modeling real-world energy systems where renewable integration introduces significant uncertainties.

Core claim

The authors construct an Ito stochastic differential equation system for regional demand, external supply, imports, and renewable integration, with multiplicative noise terms. They establish the global existence of unique nonnegative solutions, prove moment boundedness and persistence, and derive matrix inequality conditions ensuring almost sure exponential stability. Numerical schemes like Euler-Maruyama confirm that stochastic perturbations alter capacity compared to the deterministic case but preserve key properties under suitable conditions.

What carries the argument

Multiplicative Ito noise terms that scale with state variables to preserve nonnegativity and enable positivity proofs for the energy system states.

Load-bearing premise

The noise is assumed to be multiplicative, meaning fluctuations are proportional to current state levels, which is essential for keeping solutions nonnegative and for the boundedness proofs.

What would settle it

Simulation or observation showing that solutions become negative or unbounded when the noise intensity exceeds the derived bounds, or when additive noise is used instead.

Figures

Figures reproduced from arXiv: 2605.02894 by I. D. Ezekiel, L. Hairong, S. Kaennakham, S.Noeiaghdam, S.O.Edeki.

Figure 1
Figure 1. Figure 1: Graphical representation of the supply demand model [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stochastic Trajectory of External Demand ( [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stochastic Trajectory of External Supply ( [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Stochastic Trajectory of Imported Energy ( [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Stochastic Trajectory of Renewable Energy ( [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Strong error convergence of the Euler–Maruyama and Milstein schemes. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sensitivity of long-term average demand. [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

In this work, a stochastic energy supply-demand model with renewable integration is developed and analyzed. The basic nonlinear deterministic model describing the relationship among regional demand, external supply, energy imports, and renewable resource integration is extended to an Ito-type stochastic system that captures the uncertainties due to market volatility, climatic variation, policy interventions, and technical changes. Also, the noise structure is multiplicative, ensuring proportional fluctuations and preservation of nonnegativity of the state variables. Global existence, uniqueness of positive solutions, moment boundedness, and stochastic persistence are established rigorously. Furthermore, the deterministic system is analyzed, and stochastic stability is examined using matrix inequality criteria to guarantee almost sure exponential stability of the system in the stochastic setting. Among other results, stochastic perturbations significantly alter the effective system capacity compared to the deterministic case; however, under suitable parameter conditions, boundedness and stability cases are preserved. The Euler-Maruyama scheme is employed to perform numerical simulations to illustrate various dynamical behaviors and highlight the effects of uncertainty on system dynamics.The numerical reliability of the proposed model is further confirmed by additional numerical experiments via the Milstein scheme and parameter sensitivity analysis. Moreover, the results indicate that stochastic effects should be considered for capturing complex energy systems' behavior under uncertainty and its implications for renewable integration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops an Ito SDE extension of a nonlinear deterministic energy supply-demand model that incorporates regional demand, external supply, imports, and renewable integration. Multiplicative noise is used to preserve nonnegativity. The authors claim rigorous proofs of global existence and uniqueness of positive solutions, moment boundedness, and stochastic persistence. The deterministic system is analyzed separately, and almost-sure exponential stability is asserted via matrix inequality criteria. Numerical illustrations employ the Euler-Maruyama scheme, with additional checks via the Milstein scheme and parameter sensitivity analysis.

Significance. If the global stability claims hold for the full nonlinear drift, the work supplies a mathematically rigorous stochastic framework for energy systems under market, climatic, and policy uncertainty. Such results could inform capacity planning for renewables, provided the matrix criteria are shown to control the nonlinear terms globally rather than locally.

major comments (1)
  1. [Stability analysis (matrix inequality criteria)] The abstract and stability section assert almost-sure exponential stability via matrix inequality criteria, yet the underlying drift is nonlinear (demand-supply-renewable coupling). Standard LMI or Lyapunov-matrix conditions are typically derived for linear or locally linearized SDEs; the manuscript must explicitly construct a global Lyapunov function or generator inequality that absorbs the higher-order nonlinear terms and confirms negative definiteness remains intact. Without this step the global a.s. stability claim is not yet load-bearing.
minor comments (2)
  1. [Results and discussion] The abstract states that stochastic perturbations alter effective system capacity, but no explicit comparison (e.g., equilibrium shift or moment bounds) between deterministic and stochastic equilibria is quantified in the provided text; a table or corollary making this comparison precise would improve clarity.
  2. [Model formulation] Notation for the state variables (demand, supply, renewable fraction) and the precise form of the multiplicative diffusion coefficients should be introduced once in a dedicated model-equation subsection rather than scattered across the introduction and analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will revise the paper to strengthen the presentation of the stability result.

read point-by-point responses

  1. Referee: The abstract and stability section assert almost-sure exponential stability via matrix inequality criteria, yet the underlying drift is nonlinear (demand-supply-renewable coupling). Standard LMI or Lyapunov-matrix conditions are typically derived for linear or locally linearized SDEs; the manuscript must explicitly construct a global Lyapunov function or generator inequality that absorbs the higher-order nonlinear terms and confirms negative definiteness remains intact. Without this step the global a.s. stability claim is not yet load-bearing.

    Authors: We agree that the derivation of the matrix inequality criteria must be presented more explicitly to confirm it controls the full nonlinear drift globally. In the current manuscript the criteria arise from applying the infinitesimal generator to the quadratic Lyapunov function V(x)=x^TPx and using the moment-boundedness result (Theorem 3.2) together with Young's inequality to dominate the higher-order coupling terms. Nevertheless, to make this step fully transparent we will expand the stability section with a detailed computation of LV(x) that isolates each nonlinear term, shows how it is absorbed under the stated matrix condition, and verifies that the resulting inequality LV(x)≤−λ|x|^2 holds for all x in the positive orthant. The revised version will therefore contain an explicit global generator inequality rather than relying on the matrix condition alone. We will also add a short remark in the abstract clarifying that the stability holds for the nonlinear system. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard SDE theory applied directly

full rationale

The paper develops a stochastic extension of a nonlinear deterministic energy model and applies standard Ito SDE results for global existence/uniqueness of positive solutions, moment boundedness, and persistence. Stochastic stability is obtained via matrix inequality criteria on the stated drift and diffusion coefficients. No self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or ansatzes imported via prior work appear in the abstract or described derivation. The Euler-Maruyama and Milstein schemes are standard numerical methods used only for illustration, not for establishing the analytic claims. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the model is described only at the level of 'basic nonlinear deterministic model extended to Ito-type stochastic system' with multiplicative noise. No fitted constants or new postulated quantities are named.

pith-pipeline@v0.9.0 · 5544 in / 1249 out tokens · 50460 ms · 2026-05-15T15:07:14.347106+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    H. Su, E. Zio, J. Zhang, Z. Li, H. Wang, F. Zhang, and W. Wang, A systematic method for the analysis of energy supply reliability in complex integrated energy systems considering uncertainties of renewable energies, demands and operations,Journal of Cleaner Production, 267 (2020), 122117

    work page 2020

  2. [2]

    Z. Zhao, N. Holland, and J. Nelson, Optimizing smart grid performance: A stochastic approach to renewable energy integration,Sustainable Cities and Society, 111 (2024), 105533

    work page 2024

  3. [3]

    A. F. Naghibi, E. Akbari, S. Shahmoradi, S. Pirouzi, and A. Shahbazi, Stochastic economic sizing and placement of renewable integrated energy system with combined hydrogen and power technology in the active distribution network,Scientific Reports, 14(1) (2024), 28354

    work page 2024

  4. [4]

    Zhang, C

    Y. Zhang, C. Cheng, H. Cai, X. Jin, Z. Jia, X. Wu, and T. Yang, Long-term stochastic model predictive control and efficiency assessment for hydro-wind-solar renewable energy supply sys- tem,Applied Energy, 316 (2022), 119134

    work page 2022

  5. [5]

    Muhammad, E

    A. Muhammad, E. Jeroh, Y. I. Nuhu, M. E. Elton, and M. B. I. Mike, Mathematical modeling of dynamic interactions in green energy transition for economic and environmental sustain- ability,J. Environ. Sci. Econ., 2 (2023), 37–49

    work page 2023

  6. [6]

    Di Persio, M

    L. Di Persio, M. Alruqimi, and M. Garbelli, Stochastic approaches to energy markets: From stochastic differential equations to mean field games and neural network modeling,Energies, 17(23) (2024), 6106

    work page 2024

  7. [7]

    G. G. Dranka, P. Ferreira, and A. I. F. Vaz, Integrating supply and demand-side management in renewable-based energy systems,Energy, 232 (2021), 120978

    work page 2021

  8. [8]

    Cheng, M

    L. Cheng, M. Li, C. Tan, P. Huang, M. Zhang, and R. Sun, Computational game-theoretic models for adaptive urban energy systems: A comprehensive review,Archives of Computational Methods in Engineering, (2025), 1–78

    work page 2025

  9. [9]

    Ibrahim, A minimal deterministic model reveals integration of spindle assembly and position checkpoints in mitosis,Scientific Reports, 15(1) (2025), 26677

    B. Ibrahim, A minimal deterministic model reveals integration of spindle assembly and position checkpoints in mitosis,Scientific Reports, 15(1) (2025), 26677. 22

    work page 2025

  10. [10]

    Daneshvar, B

    M. Daneshvar, B. Mohammadi-Ivatloo, K. Zare, and S. Asadi, Two-stage robust stochastic model scheduling for transactive energy based renewable microgrids,IEEE Transactions on Industrial Informatics, 16(11) (2020), 6857–6867

    work page 2020

  11. [11]

    Cheng, C

    S. Cheng, C. Quilodr´ an-Casas, S. Ouala, A. Farchi, C. Liu, P. Tandeo, and R. Arcucci, Ma- chine learning with data assimilation and uncertainty quantification for dynamical systems: A review,IEEE/CAA Journal of Automatica Sinica, 10(6) (2023), 1361–1387

    work page 2023

  12. [12]

    Jafari, T

    A. Jafari, T. Khalili, H. G. Ganjehlou, and A. Bidram, Optimal integration of renewable energy sources, diesel generators, and demand response program,Journal of Cleaner Production, 247 (2020), 119100

    work page 2020

  13. [13]

    Z. Lai, W. Liu, X. Jian, K. Bacsa, L. Sun, and E. Chatzi, Neural modal ordinary differential equations,Data-Centric Engineering, 3 (2022), e34

    work page 2022

  14. [14]

    Chebabhi, I

    A. Chebabhi, I. Tegani, and A. D. Benhamadouche, Modeling and forecasting uncertainties in renewable energy system,Computers and Electrical Engineering, 127 (2025), 110588

    work page 2025

  15. [15]

    D. Wang, H. Dai, Y. Jin, Z. Li, S. Luo, and X. Li, Employing quantum entanglement for real- time coordination of distributed electric vehicle charging stations,Energies, 18(11) (2025), 2917

    work page 2025

  16. [16]

    Zhong, X

    Z. Zhong, X. Fang, J. Li, Q. Ma, R. Zhou, Y. Hu, and S. Du, Linear and non-linear dynamics of ecosystem services supply and demand,Ecological Indicators, 158 (2024), 111614

    work page 2024

  17. [17]

    J. C. Do Prado and W. Qiao, A stochastic bilevel model for an electricity retailer,IEEE Transactions on Sustainable Energy, 11(4) (2020), 2803–2812

    work page 2020

  18. [18]

    Q. Li, P. Liu, X. Meng, G. Zhang, Y. Ai, and W. Chen, Model prediction control-based energy management,IEEE Transactions on Transportation Electrification, 8(2) (2022), 2249–2260

    work page 2022

  19. [19]

    J. T. Rovimar, Extremum seeking for the first derivative of nonlinear maps with constant delays,European Journal of Emerging Engineering and Mathematics, 1(01) (2024), 34–49

    work page 2024

  20. [20]

    Zhuang and H

    P. Zhuang and H. Liang, Stochastic energy management of electric bus charging stations,IEEE Transactions on Sustainable Energy, 12(2) (2020), 1206–1216

    work page 2020

  21. [21]

    M. A. Bozkurt, Y. K¨ ose, and S. C ¸ elik, Analyzing the dynamic behavior and market efficiency of green energy investments,Energy Sources, Part B, 20(1) (2025), 2457438

    work page 2025

  22. [22]

    B. Yan, M. Di Somma, G. Graditi, and P. B. Luh, Markovian-based stochastic operation optimization,Electric Power Systems Research, 186 (2020), 106364

    work page 2020

  23. [23]

    Ogundile and S

    O. Ogundile and S. Edeki, Semi-analytical methods for solving Itˆ o stochastic models,Modeling and Computation in Vibration Problems, 2 (2021), 11–1. 23

    work page 2021

  24. [24]

    M. Wang, H. Yu, Y. Yang, X. Lin, H. Guo, C. Li, and R. Jing, Unlocking emerging impacts of carbon tax on integrated energy systems,Applied Energy, 302 (2021), 117579

    work page 2021

  25. [25]

    O. P. Ogundile and S. O. Edeki, Karhunen–Lo´ eve expansion of Brownian motion,J. Math. Comput. Sci., 10(5) (2020), 1712–1723

    work page 2020

  26. [26]

    J. Wang, X. Qi, F. Ren, G. Zhang, and J. Wang, Optimal design of hybrid combined cooling, heating and power systems,Journal of Cleaner Production, 281 (2021), 125357

    work page 2021

  27. [27]

    W. Gan, M. Yan, W. Yao, J. Guo, J. Fang, X. Ai, and J. Wen, Multi-network coordinated hydrogen supply infrastructure planning,IEEE Transactions on Industry Applications, 58(2) (2021), 2875–2886

    work page 2021

  28. [28]

    Lamba, J

    H. Lamba, J. C. Mattingly, and A. M. Stuart, An adaptive Euler–Maruyama scheme for SDEs, IMA Journal of Numerical Analysis, 27(3) (2007), 479–506

    work page 2007

  29. [29]

    Abunima, W

    H. Abunima, W. H. Park, M. B. Glick, and Y. S. Kim, Two-stage stochastic optimization for operating a renewable-based microgrid,Applied Energy, 325 (2022), 119848

    work page 2022

  30. [30]

    S. E. Rahman, Euler–Maruyama’s method for numerical solution of Ornstein–Uhlenbeck’s equation, Undergraduate thesis, Padang State University (2020), 20–21

    work page 2020

  31. [31]

    Tatari and H

    F. Tatari and H. Modares, Deterministic and stochastic fixed-time stability of discrete-time autonomous systems,IEEE/CAA Journal of Automatica Sinica, 10(4) (2023), 945–956

    work page 2023

  32. [32]

    Boyko and N

    V. Boyko and N. Vercauteren, A stochastic stability equation for unsteady turbulence,Quar- terly Journal of the Royal Meteorological Society, 149(755) (2023), 2125–2145

    work page 2023

  33. [33]

    E. Gul, G. Baldinelli, A. Farooqui, P. Bartocci, and T. Shamim, AEM-electrolyzer based hydrogen integrated renewable energy system optimisation model,Energy Conversion and Management, 285 (2023), 117025

    work page 2023

  34. [34]

    Noeiaghdam and D

    S. Noeiaghdam and D. Sidorov, Caputo–Fabrizio fractional derivative to solve the fractional model of energy supply-demand system,Mathematical Modelling of Engineering Problems, 7(3) (2020), 359–367

    work page 2020

  35. [35]

    Liang, J

    J. Liang, J. Miao, L. Sun, L. Zhao, J. Wu, P. Du, and W. Zhao, Supply–demand dynamics quantification and distributionally robust scheduling,Energies, 18(5) (2025), 1181

    work page 2025

  36. [36]

    V. T. Vo, S. Noeiaghdam, A. I. Dreglea, and D. N. Sidorov, A study of numerical methods for solving the nonlinear energy resources supply-demand system,Middle Volga Mathematical Society Journal, 27(2) (2025), 143–170

    work page 2025

  37. [37]

    X. Ren, J. Li, F. He, and B. Lucey, Impact of climate policy uncertainty on traditional energy and green markets,Renewable and Sustainable Energy Reviews, 173 (2023), 113058. 24

    work page 2023

  38. [38]

    Oyekan, E

    M. Oyekan, E. Igba, and S. O. Jinadu, Building resilient renewable infrastructure in an era of climate and market volatility,International Journal of Scientific Research in Humanities and Social Sciences, 1(1) (2024), 217–242. 25

    work page 2024