arxiv: 2605.12405 · v1 · submitted 2026-05-12 · 🧮 math-ph · math.MP· math.PR· physics.data-an

Recognition: 2 theorem links

· Lean Theorem

An analytical approach to calculating stationary PDFs for reflected random walks with an application to BESS-based ramp-rate control

\'Alvaro Herrera, Carlos Colchero, Diego Jim\'enez-Arregu\'in, Jorge E. P\'erez-Garc\'ia, Oliver Probst

Pith reviewed 2026-05-13 03:20 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PRphysics.data-an

keywords reflected random walksstationary PDFWiener-Hopf integral equationNeumann seriesBESSramp-rate controlinverter sizingrenewable energy

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The pith

An exact analytical solution for the stationary PDF of a reflected random walk is derived from a Wiener-Hopf integral equation using a Neumann series.

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

The paper derives a rigorous integral equation for the stationary distribution of a reflected random walk using probability theory. It then solves this equation analytically with a Neumann series expansion. This approach is applied to modeling battery energy storage systems used for controlling the rate of change in power from wind and solar sources. The analytical results allow for precise calculations of required inverter sizes without relying solely on simulations. Practitioners can use truncated versions of the solution to create simple design guidelines for such systems.

Core claim

The central claim is that the stationary PDF of the reflected random walk can be found by solving a derived Wiener-Hopf-type integral equation with an analytical Neumann series solution, which is then used to analyze BESS power for ramp-rate control.

What carries the argument

The Wiener-Hopf-type integral equation for the stationary PDF solved by Neumann series.

If this is right

The analytical solution agrees with numerical Nystrom method and simulation results.
Truncated versions yield simplified design rules for BESS inverter sizing.
General insights into sizing criteria for storage systems in VRE ramp-rate control are obtained.
The method provides an alternative to pure numerical or simulation-based approaches.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be used to derive sensitivity of the stationary distribution to model parameters for better optimization of BESS.
Similar methods might apply to other Markov processes with reflection in engineering applications.
Exact analytical forms enable probabilistic assessments of ramp compliance in renewable systems.

Load-bearing premise

The BESS power dynamics are accurately captured by the chosen reflected random walk Markov kernel whose transition probabilities are independent of the stationary distribution.

What would settle it

Significant discrepancy between the Neumann series results and those from the Nystrom method or from direct simulation of the input time series would falsify the analytical solution.

Figures

Figures reproduced from arXiv: 2605.12405 by \'Alvaro Herrera, Carlos Colchero, Diego Jim\'enez-Arregu\'in, Jorge E. P\'erez-Garc\'ia, Oliver Probst.

**Figure 2.** Figure 2: Performance of the analytical solution against the algorithmic battery simulation. [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: Validation of g˜( ˜b, a˜) independence on a and βSL. (a) Comparison of P99(a˜) values for 3 different cases along the a˜ domain. (b) Equivalence of each case following a˜ = a1β2 = a2β1 = a3β3. research question (3): Considering that the analytical solution has been stated in terms of an infinite power series, what are the trade-offs in terms of accuracy vs. computation time? To answer this question, we fir… view at source ↗

**Figure 4.** Figure 4: a) Comparison of the algorithmic solution using [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

**Figure 5.** Figure 5: GL influence in the BESS PDF for different [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Impact of weight ratio parameter c on GL-distribution power changes for BESS PDF (colored lines) showing the P99 curve comparison against the SL-analytical Neumann series solution (black line). 3.4. Sizing of BESS inverters Having found the BESS power PDF from the solution to the integral equation using Neumann series, the required BESS power capacity can be determined easily in terms of a suitable P-value… view at source ↗

read the original abstract

A Wiener-Hopf-type integral equation for the stationary PDF of a reflected random walk is derived rigorously based on modern probability theory, and an application to battery energy storage systems (BESS), specifically the sizing of the inverter, is discussed in depth. The methodological steps include the construction of a Markov kernel, the derivation of a Fredholm integral equation of the second kind for the PDF of the BESS power, and an analytical solution of the equation based on a Neumann series. The analytical results were compared against numerical solutions obtained with the Nystrom method, as well as against the results of an algorithmic simulation using simulated input time series. The use of truncated versions of the analytic solution allows for the construction of simplified design rules for the power systems practitioner. General insights into inverter sizing criteria of storage systems for ramp-rate control of variable renewable energy (VRE) sources such as wind and solar are provided.

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.

Desk Editor's Note private letter to a colleague

The paper derives a Neumann-series solution for the stationary PDF of a reflected random walk and applies truncated versions to give practical BESS inverter sizing rules.

read the letter

The key point is that this paper derives a Neumann series for the stationary probability density of a reflected random walk starting from a Wiener-Hopf integral equation, and then uses it to produce practical sizing guidelines for battery inverters in ramp-rate control. They start with a Markov kernel built from first principles for the power output process, convert the stationarity condition into a Fredholm integral equation of the second kind, and expand the solution in a Neumann series. The resulting expression is then checked against both a numerical Nyström solver and direct simulations on generated time series. Truncating the series gives simple closed-form approximations that can be turned into design rules without heavy computation. That analytic route is the main advance over pure simulation approaches that dominate current practice in this area. The modeling choice for the kernel is the main soft spot. The transition probabilities are fixed in advance and do not depend on the stationary distribution itself. If the actual BESS control introduces feedback that depends on the current state or if the renewable generation has temporal correlations that affect the effective transitions, the predicted PDF could deviate from reality. The paper reports comparisons but does not include specific error metrics, so the accuracy remains somewhat qualitative. This is useful for engineers and modelers working on storage integration for wind and solar. Readers who need quick analytic estimates for similar reflected processes will get direct value from the method. It is solid enough on the derivation and internal checks to merit a full referee process. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper derives a Wiener-Hopf-type integral equation for the stationary PDF of a reflected random walk from a Markov kernel constructed via modern probability theory, solves the resulting Fredholm equation of the second kind analytically using a Neumann series, and applies the truncated series to obtain simplified design rules for inverter sizing in BESS-based ramp-rate control of VRE sources. Analytical results are compared to Nyström numerical solutions and algorithmic simulations on simulated time series.

Significance. If the derivation and modeling assumptions hold, the work supplies an analytical framework for stationary distributions of reflected random walks that can yield practical, truncated closed-form approximations for power-system design. The internal consistency checks against Nyström methods and direct simulation are a clear strength, as is the explicit construction of the Markov kernel from first principles without fitted parameters. The BESS application illustrates how such expressions translate into inverter sizing criteria, though the utility depends on the fidelity of the chosen kernel to real inverter dynamics and VRE correlations.

major comments (2)

[BESS application and Markov kernel construction] The central modeling step for the BESS application (described in the abstract and the application section) fixes the transition probabilities of the reflected random walk independently of the unknown stationary distribution. Real inverter control feedback loops or temporal correlations in VRE inputs can render the effective kernel occupancy-dependent; if this occurs, the stationary PDF obtained from the integral equation will not match the actual power distribution, undermining the derived design rules. A concrete justification or sensitivity analysis of this independence assumption is required.
[Validation and results] The abstract states that analytic results were compared against Nyström solutions and algorithmic simulations, yet supplies no quantitative error metrics (e.g., integrated absolute error, maximum pointwise deviation, or convergence rates with respect to truncation order). Without these, it is impossible to assess how well the Neumann series approximates the true stationary PDF or to judge the practical accuracy of the truncated design rules.

minor comments (2)

[Derivation] Notation for the reflected random walk state space and the precise definition of the reflection mechanism should be stated explicitly at the outset of the derivation section to avoid ambiguity when the Wiener-Hopf equation is introduced.
[Neumann series solution] The manuscript would benefit from a brief remark on the radius of convergence of the Neumann series or on conditions guaranteeing its applicability to the specific kernel arising from the BESS model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: The central modeling step for the BESS application (described in the abstract and the application section) fixes the transition probabilities of the reflected random walk independently of the unknown stationary distribution. Real inverter control feedback loops or temporal correlations in VRE inputs can render the effective kernel occupancy-dependent; if this occurs, the stationary PDF obtained from the integral equation will not match the actual power distribution, undermining the derived design rules. A concrete justification or sensitivity analysis of this independence assumption is required.

Authors: We appreciate the referee highlighting this modeling choice. The Markov kernel is constructed from the deterministic ramp-rate control law and the known statistical properties of VRE increments, so that transition probabilities are independent of the unknown stationary PDF by design; the integral equation then yields the self-consistent stationary distribution. This independence is a deliberate simplification that enables the closed-form Neumann-series solution. We acknowledge that real-world feedback or strong temporal correlations could introduce occupancy dependence. In the revised manuscript we will add a paragraph in Section 3.2 explicitly justifying the assumption from the control architecture and include a sensitivity study in which the kernel parameters are perturbed to quantify the effect on the resulting design rules. revision: partial
Referee: The abstract states that analytic results were compared against Nyström solutions and algorithmic simulations, yet supplies no quantitative error metrics (e.g., integrated absolute error, maximum pointwise deviation, or convergence rates with respect to truncation order). Without these, it is impossible to assess how well the Neumann series approximates the true stationary PDF or to judge the practical accuracy of the truncated design rules.

Authors: We agree that quantitative error metrics are necessary for a rigorous assessment. Although visual comparisons appear in Section 4, explicit numerical measures are not reported. In the revised version we will insert a new table (and accompanying text) that tabulates the integrated absolute error, maximum pointwise deviation, and convergence rate of the truncated Neumann series as a function of truncation order N, for both the Nyström reference solution and the direct Monte-Carlo simulations. These additions will allow readers to judge the practical accuracy of the truncated design rules. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from first-principles kernel

full rationale

The paper constructs a Markov kernel for the reflected random walk from modeling assumptions on BESS dynamics, derives a Wiener-Hopf/Fredholm integral equation of the second kind for the stationary PDF, and obtains an analytical Neumann-series solution that is then truncated for design rules. This chain is compared to independent Nystrom numerics and Monte-Carlo simulation on input time series. No quoted step reduces the final PDF expression to a fitted parameter, self-citation, or redefinition of the input kernel; the stationary distribution is the standard output of solving the balance equation given a fixed transition kernel. The modeling choice of kernel independence from the unknown PDF is the usual Markov assumption and does not constitute circularity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and uniqueness of a stationary distribution for the reflected random walk and on the validity of the Markov kernel as a model for BESS power; no free parameters are explicitly named in the abstract, and no new physical entities are introduced.

axioms (2)

domain assumption The reflected random walk possesses a unique stationary probability distribution.
Invoked when the stationary PDF is defined as the solution of the integral equation.
standard math The transition kernel of the walk can be written explicitly from the underlying stochastic process without reference to the stationary measure.
Required to construct the Fredholm equation of the second kind.

pith-pipeline@v0.9.0 · 5490 in / 1454 out tokens · 68373 ms · 2026-05-13T03:20:52.967077+00:00 · methodology

discussion (0)

Reference graph

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