An Analytical Probabilistic Expression for Modeling Sum of Spatial-dependent Wind Power Output

Libao Shi; Yang Pan; Yixin Ni

REVIEW 4 major objections 6 minor 8 references

An Analytical Probabilistic Expression for Modeling Sum of Spatial-dependent Wind Power Output

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The total output of two spatially dependent wind farms has a closed-form probability density, including the discrete masses at zero and rated power, and it matches Monte Carlo simulation while running much faster.

desk verdict Honest incremental progress on a two-farm wind power sum distribution, but the undisclosed GMM fit and the miscounted N-farm extension keep it from being reproducible as written. read the letter →

arxiv 1908.05817 v1 pith:V3PF26KZ submitted 2019-08-16 eess.SY cs.SY

Libao Shi , Yang Pan , Yixin Ni This is my paper · ORCID

classification eess.SYcs.SY

keywords windpowerprobabilitydensityfunctioncopulaunitimpulseGaussianmixturemodelsumofdependentrandomvariablesMonteCarlovalidationspatialdependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the probability density of the total power output from two spatially dependent wind farms can be written as a closed-form analytical expression, rather than obtained by repeated numerical simulation. The difficulty is that each farm's output carries discrete probability masses at zero and at rated power, so the sum's density has Dirac impulses at four terminals. The paper builds the density by combining copula-based joint distributions with a diagonal-difference method for corner probabilities and a Gaussian mixture fit for the continuous parts, and states that the resulting curves match Monte Carlo simulation while using substantially less computation. If correct, the formula gives a fast, accurate building block for stochastic power-system analysis in which many wind farms' outputs must be summed.

What carries the argument

The load-bearing construction is the split of the sum's CDF into a region integral and a corner term, together with equation (7)'s piecewise assembly. Sklar's theorem converts the joint density into $c(W_1(P_{wf1}),W_2(P_{wf2}))\,w_1(P_{wf1})\,w_2(P_{wf2})$; the unit impulse functions represent the discrete atoms at $P_{wfi}=0$ and $P_{wfi}=P_{wfri}$; the diagonal-difference method computes the corner probability from the joint CDF via expressions like $\Phi_1(0)=M(0,0)$ and $\Phi_2(P_{wfr1})=M(P_{wfr1},0)-M(P_{wfr1}^-,0)$; and a Gaussian mixture model fits the part of the integrand that lacks a closed form. Working with the CDF first avoids the difficulty of integrating Dirac impulses directly in the convolution integral.

What would settle it

Use two wind farms with exactly known marginal distributions and a copula for which the convolution can be evaluated by high-precision numerical quadrature, then compare each $\Phi_i$ and $\phi_i$ in equation (7) with that benchmark at the four terminals and on the three open intervals. Any mismatch in an impulse size or a continuous segment would locate the error in the diagonal-difference corner terms or the Gaussian mixture fit.

Watch

Extended reading notes

Core claim

The central claim is equation (7), the final analytical expression of the PDF $w(P_{wf})$ for $P_{wf}=P_{wf1}+P_{wf2}$. It is piecewise defined: continuous parts $\phi_1$, $\phi_2$, $\phi_3$ on the intervals between the terminals $0$, $P_{wfr1}$, $P_{wfr2}$, and $P_{wfr1}+P_{wfr2}$, plus four Dirac impulses at those terminals with sizes $\Phi_1(0)=M(0,0)$, $\Phi_2(P_{wfr1})=M(P_{wfr1},0)-M(P_{wfr1}^-,0)$, $\Phi_3(P_{wfr2})=M(0,P_{wfr2})-M(0,P_{wfr2}^-)$, and $\Phi_4(P_{wfr1}+P_{wfr2})=1-M(P_{wfr1}^-,P_{wfr2})-M(P_{wfr1},P_{wfr2}^-)+M(P_{wfr1}^-,P_{wfr2}^-)$. The derivation first finds the CDF as a region integral plus a corner probability obtained by the diagonal-difference method, then differentiates. The paper reports that in the two-farm example the model and Monte Carlo curves are perfectly matched, with CPU time dropping from 22.046 s at 200,000 samples to 0.5208 s for the proposed expression.

Load-bearing premise

The derivation rests on the marginal PDFs for each wind farm—taken from earlier work—being correct, including their representation of zero and rated output as unit impulse atoms; if those marginals are wrong, equation (7) collapses.

Editorial extensions

If this is right

The total-output PDF for two dependent farms is available in closed form, so probabilistic power-flow and stochastic dispatch calculations can avoid repeated Monte Carlo sampling.
The same region-division and corner-probability logic extends to $N$ farms, with $2N$ critical points and $2N-1$ segments.
The expression works with any copula family and any marginal PDFs of the assumed form, so the method can be re-used as long as the marginals are known.
The four Dirac impulse sizes give exact probability masses at zero, rated, and total-rated output, quantities that matter for curtailment and reliability studies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not quantify the approximation error introduced by the Gaussian mixture fit on the continuous segments; a reader should treat 'analytical' as 'semi-analytical with a controlled curve fit' until an error bound is supplied.
Although the extension to $N$ farms is sketched, the corner combinations grow combinatorially, so the practical advantage over Monte Carlo likely shrinks as $N$ grows; testing at $N=3$ or $4$ would show where the method's speed benefit ends.
A natural next test is to embed the resulting PDF into a chance-constrained dispatch or probabilistic load flow and compare solution time and accuracy against a Monte-Carlo-based benchmark; the paper demonstrates the density-level speedup but not the optimization-level payoff.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is honest incremental progress, not a breakthrough. The two-farm case gives a piecewise PDF for the sum with Dirac impulses at the terminals, which I don't think is in the prior copula convolution literature. The CDF/diagonal-difference route for corner probabilities is the right instinct and the impulse-size formulas in (8) look plausible. Credit where due: the authors correctly identify that discrete atoms at 0 and rated output need special handling, and Sklar's theorem plus convolution is a sensible skeleton.

The soft spots are real and mostly concentrated. The GMM inside (7) is the biggest one. Equation (5) says a GMM 'is applied to fit an item that is hard to be analytically expressed,' but the paper never says what is being fitted, with what order, what coefficients, or with what error. That makes eq. (7) not a uniquely defined analytical expression: different fits give different PDFs. The stress-test note is right that this is more than a presentation gap. Second, Regions II and III are asserted 'derived in a similar way' with no derivation, and the formulas (10)-(11) contain several mixed partial/CDF terms that need checking. Third, the N-farm extension mis-counts: with N farms each at 0 or rated, you have 2^N discrete corners, not 2N, and the number of critical subset sums is 2^N, not 2N. That part is wrong as written. Fourth, validation is visual; 'perfectly matched' is unsupported by any error metric or normalization check. Given the GMM approximation, the total mass of w(Pwf) should be verified.

The self-citation of [7] for the marginal PDFs is fine—it's their own prior result and the relevant one—but the marginal construction is not reproduced, so a reader has to trust [7].

Is the central argument salvageable? Yes, I think so. The two-farm derivation is a plausible construction and the corner handling via CDF differences is sound in principle. The paper reads like a compressed letter that cut the details to fit. A serious referee could push for the missing pieces. But as it stands, eq. (7) is not reproducible from the text, and the N-farm claim is simply incorrect in the enumeration. I would not cite it in its current form.

Recommendation: send it to peer review, but do not expect it to pass until the GMM target and parameters are disclosed, the two-farm branches are fully derived, and the N-farm counting is fixed. The reader's conditional verdict is fair.

Referee Report

4 major / 6 minor

Summary. The manuscript develops a piecewise 'analytical probabilistic expression' for the probability density function (PDF) of the sum of two spatially dependent wind farm outputs. The derivation combines marginal PDFs with unit impulses at zero and rated output (taken from the authors' prior work [7]), a copula to model dependence via Sklar's theorem, and a Gaussian mixture model (GMM) to handle a term that lacks a closed form. The final expression, Eq. (7), consists of four Dirac impulses at discrete total-output values and three continuous branches for the intervals between them. The paper then sketches an extension to N wind farms and presents a Monte Carlo comparison for one two-farm test case.

Significance. If the proposed expression were fully defined, correct, and validated, it would offer a fast closed-form alternative to Monte Carlo simulation for the sum of dependent wind farm outputs, which is of practical interest in probabilistic power system analysis. The paper's high-level idea of combining copula-based convolution with an explicit treatment of probability atoms at zero and rated output is reasonable, and the Region I derivation illustrates a sensible decomposition into a continuous integral and a diagonal-difference boundary term. The CPU-time comparison is also a useful concrete demonstration. However, as written the central claim is not supported: Eq. (7) is not a uniquely specified expression because the GMM parameters are omitted, the N-farm extension miscounts the possible atom locations, two of the three continuous branches are asserted without derivation, and the validation is purely qualitative.

major comments (4)

[Section II-A, Eqs. (5), (7), (9)-(11)] The Gaussian mixture model is not specified. The text states that the GMM 'as shown in (5)' is applied to fit an item that is hard to express analytically, but no order n, no coefficients a_i, b_i, c_i, no fitting target (the integrand, the convolution integral, or some intermediate factor), and no fitting procedure are given. As a result, Eq. (7) is not a uniquely defined expression and cannot be reproduced by a reader. Since the GMM approximation enters the continuous branches (9)-(11), the phrase 'analytical probabilistic expression' is an overstatement, and no error bound or normalization check is provided to quantify the resulting approximation error.
[Section II-B, N-farm extension] The extension to N wind farms mis-enumerates the critical points. The paper states that there are '2 N critical points' and lists 0, the individual rated outputs Pwfr1,...,PwfrN, and the cumulative sums Pwfr1+Pwfr2,...,Pwfr1+...+PwfrN. The actual set of possible atom locations for the sum of N farms is the set of all 2^N subset sums of the rated outputs; for N=3, for example, the subset sums Pwfr1+Pwfr3 and Pwfr2+Pwfr3 are missing from the list. Consequently, the proposed segmentation into 2N-1 intervals does not cover the piecewise structure, and the claimed expansion is not valid as stated.
[Section II-A, Eqs. (10)-(11)] The formulas for Regions II and III are stated without derivation. The paper says only that 'the PDFs in Regions II and III can be derived in a similar way,' but the boundary terms, partial derivatives of the copula, and integration limits in Eq. (10) and Eq. (11) are nontrivial and are not self-evidently correct. Because these two branches are part of the central piecewise expression (7), the expression is not established for two of its three continuous branches without a derivation or a detailed consistency check.
[Section III, Fig. 2 and Table I] The validation is qualitative and limited to a single parameter setting (Weibull and Gumbel margins with a Gumbel copula, alpha=3.65). Fig. 2 provides no numerical error metric and no normalization check for the proposed PDF, so the claim that the curves are 'perfectly matched' is not quantitatively supported. In addition, the N-farm extension is not validated at all, despite being presented as part of the contribution.

minor comments (6)

[Section II-A, equation numbering] Equation (5) is used for both the GMM sum and for WII(Pwf), and the text states that 'The detailed CDF expression WII(Pwf) is given in (5)' immediately after displaying the GMM sum; the equations should be renumbered to avoid ambiguity.
[Section II-B, notation] The expression '2*N corners: {Pwf1,Pwf2,...,Pwfi,...,PwfN, |Pwfi=0 or Pwfi=Pwfri}' is ambiguous and not the standard notion of a corner of an N-dimensional rectangle; a corner should be a vector in which every coordinate is either 0 or rated.
[Section II-A, marginal PDFs] The derivation depends on the marginal PDFs w1 and w2 from reference [7], but these functions are not reproduced or summarized; a reader cannot implement Eq. (7) without consulting that prior work.
[Section III, Table I] Table I should clarify how the CPU time for the proposed model was measured and whether it is independent of the Monte Carlo sample size; the table layout is also difficult to parse.
[Abstract and Introduction] There are several grammatical errors, including 'we develops' in the Introduction and 'This letter proposed analytical probabilistic expression' in Section IV; these should be corrected.
[Fig. 1 caption] The caption for Region I contains the typo 'Pwfr1+Pwfr1'; it should presumably read 'Pwfr1+Pwfr2'.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: Eq. (7) follows from convolution and Sklar's theorem; the self-cited marginals and undisclosed GMM are reproducibility concerns, not reductions to inputs.

full rationale

The derivation of Eq. (7) starts from the standard convolution identity (1), which is the definition of the PDF of a sum, and then applies Sklar's theorem (2) to express the joint density through the copula and marginal densities. These are identities and theorems, not restatements of the target result. The impulse coefficients in (8) are computed directly from the assumed joint CDF M and marginals W1, W2; they do not presuppose the final sum PDF. The continuous branches (9)-(11) include a term that the paper states is fitted by a Gaussian mixture model 'to fit an item that is hard to be analytically expressed in theory,' but the text does not say or imply that the GMM was fitted to the Monte Carlo output used for validation. The absence of GMM order, coefficients, and error bounds is a reproducibility and verification gap, not a circular reduction. The marginal PDFs w1 and w2 are taken from the authors' prior work [7]; this is a self-citation, but it supplies input wind-speed-to-power PDFs that are externally falsifiable and not identical to the derived sum distribution. Thus no equation or fitted parameter in the paper reduces by construction to its own inputs; the circularity concern is limited to reliance on self-cited inputs and an undisclosed fitted approximation, neither of which makes the central claim circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard copula theory, the prior single-farm result [7], and an undisclosed GMM fit. The N-farm extension introduces an incorrect enumeration of discrete states. No new physical entities are postulated.

free parameters (1)

GMM component parameters (n, a_i, b_i, c_i)
The paper states in Section II-A that a Gaussian mixture model, equation (5), is used to fit an item that is hard to express analytically, but the number of components and all coefficients are not reported. This is an undisclosed fitted element inside the claimed analytical expression.

assumptions (4)

standard math Sklar's theorem for copula decomposition of the joint distribution
Used in equation (2) to express the joint PDF as the copula density times the marginal densities; standard probability result.
domain assumption Single-farm output PDFs with Dirac impulses at 0 and rated output are valid as derived in [7]
The derivation begins with w1(Pwf1) and w2(Pwf2) taken from the authors' previous work [7]; no independent derivation or data is provided in this paper.
standard math The diagonal difference method from [8] correctly computes corner probabilities
Used to calculate WII and the impulse sizes Phi; it is an established method in probability bounds analysis, but the adaptation here is not fully derived.
ad hoc to paper For N wind farms, only 2N critical points need to be considered
Section II-B lists 2N critical points for the N-farm case, but the sum of N binary-valued rated outputs has 2^N possible combinations, so this assumption is mathematically incomplete and likely incorrect.

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Pith. "Pith review of An Analytical Probabilistic Expression for Modeling Sum of Spatial-dependent Wind Power Output." pith.science (2026). https://pith.science/paper/V3PF26KZ

@misc{pith2026190805817,
  author       = {Pith},
  title        = {Pith review of: An Analytical Probabilistic Expression for Modeling Sum of Spatial-dependent Wind Power Output},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3PF26KZ}},
  note         = {Machine review of arXiv:1908.05817}
}

read the original abstract

Applying probability-related knowledge to accurately explore and exploit the inherent uncertainty of wind power output is one of the key issues that need to be solved urgently in the development of smart grid. This letter develops an analytical probabilistic expression for modeling sum of spatial-dependent wind farm power output through introducing unit impulse function, copulas, and Gaussian mixture model. A comparative Monte Carlo sampling study is given to illustrate the validity of the proposed model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[7]

An analytical solution for wind farm power output,

L. B. Shi, Z. X. Weng, L. Z. Yao, and Y. X. Ni, “An analytical solution for wind farm power output,” IEEE Trans. Power Syst., vol. 29, no. 6, pp. 3122- 3123, 2014

work page 2014
[1]

Some statistical characteristics of monthly average wind speed at various heights,

K. Tar “Some statistical characteristics of monthly average wind speed at various heights,” Renewable and Sustainable Energy Reviews , vol. 12, no. 6, pp. 1712-1724, 2008

work page 2008
[2]

Statistical representation of distribution system loads using Gaussian mixture model,

R. Singh, B. C. Pal, and R.A. Jabr, “Statistical representation of distribution system loads using Gaussian mixture model,” IEEE Trans. Power Syst. , vol. 25, no. 1, pp. 29–37, Feb. 2010

work page 2010
[3]

Analytical Expressions for Joint Distributions in Probabilistic Load Flow ,

Z. W. Wang, C. Shen, F. Liu, and F. Gao, “Analytical Expressions for Joint Distributions in Probabilistic Load Flow ,” IEEE Trans. Power Syst., vol. 32, no. 3, pp. 2473-2474, 2017

work page 2017
[4]

Probabilistic load flow method based on Nataf transformation and Latin hypercube sampling,

Y. Chen, J. Wen, and S. Cheng, “Probabilistic load flow method based on Nataf transformation and Latin hypercube sampling,” IEEE Trans. Sustain. Energy, vol. 4, no. 2, pp. 294–301, Apr. 2013

work page 2013
[5]

Using copulas for modeling stochastic dependence in power system uncertainty analysis,

G. Papaefthymiou and D. Kurowicka, “Using copulas for modeling stochastic dependence in power system uncertainty analysis,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 40–49, 2009

work page 2009
[6]

Copula Based Dependent Discrete Convolution for Power System Uncertainty Analysis ,

N. Zhang, C. Q. Kang, C. Singh, and Q. Xia, “Copula Based Dependent Discrete Convolution for Power System Uncertainty Analysis ,” IEEE Trans. Power Syst., vol. 31, no. 6, pp. 5204-5205, 2016

work page 2016
[8]

Dependence in probabilistic modeling, Dempster -Shafer theory, and probability bounds analysis ,

S. Ferson, R . B. Nelson, J. Hajagos, et al. “Dependence in probabilistic modeling, Dempster -Shafer theory, and probability bounds analysis ,” Albuquerque: Sandia National Laboratories, 2004. Pwfr1 Pwfr2 Pwfr1 Region III: Pwf2<Pwf=Pwf1+Pwf2Pwfr1+Pwfr2 Region II: Pwfr1<Pwf=Pwf1+Pwf2Pwfr1+Pwfr2 Region I: 0<Pwf=Pwf1+Pwf2Pwfr1+Pwfr1 Pwfr1 Pwfr2

work page 2004

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

An analytical solution for wind farm power output,

L. B. Shi, Z. X. Weng, L. Z. Yao, and Y. X. Ni, “An analytical solution for wind farm power output,” IEEE Trans. Power Syst., vol. 29, no. 6, pp. 3122- 3123, 2014

work page 2014

[2] [1]

Some statistical characteristics of monthly average wind speed at various heights,

K. Tar “Some statistical characteristics of monthly average wind speed at various heights,” Renewable and Sustainable Energy Reviews , vol. 12, no. 6, pp. 1712-1724, 2008

work page 2008

[3] [2]

Statistical representation of distribution system loads using Gaussian mixture model,

R. Singh, B. C. Pal, and R.A. Jabr, “Statistical representation of distribution system loads using Gaussian mixture model,” IEEE Trans. Power Syst. , vol. 25, no. 1, pp. 29–37, Feb. 2010

work page 2010

[4] [3]

Analytical Expressions for Joint Distributions in Probabilistic Load Flow ,

Z. W. Wang, C. Shen, F. Liu, and F. Gao, “Analytical Expressions for Joint Distributions in Probabilistic Load Flow ,” IEEE Trans. Power Syst., vol. 32, no. 3, pp. 2473-2474, 2017

work page 2017

[5] [4]

Probabilistic load flow method based on Nataf transformation and Latin hypercube sampling,

Y. Chen, J. Wen, and S. Cheng, “Probabilistic load flow method based on Nataf transformation and Latin hypercube sampling,” IEEE Trans. Sustain. Energy, vol. 4, no. 2, pp. 294–301, Apr. 2013

work page 2013

[6] [5]

Using copulas for modeling stochastic dependence in power system uncertainty analysis,

G. Papaefthymiou and D. Kurowicka, “Using copulas for modeling stochastic dependence in power system uncertainty analysis,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 40–49, 2009

work page 2009

[7] [6]

Copula Based Dependent Discrete Convolution for Power System Uncertainty Analysis ,

N. Zhang, C. Q. Kang, C. Singh, and Q. Xia, “Copula Based Dependent Discrete Convolution for Power System Uncertainty Analysis ,” IEEE Trans. Power Syst., vol. 31, no. 6, pp. 5204-5205, 2016

work page 2016

[8] [8]

Dependence in probabilistic modeling, Dempster -Shafer theory, and probability bounds analysis ,

S. Ferson, R . B. Nelson, J. Hajagos, et al. “Dependence in probabilistic modeling, Dempster -Shafer theory, and probability bounds analysis ,” Albuquerque: Sandia National Laboratories, 2004. Pwfr1 Pwfr2 Pwfr1 Region III: Pwf2<Pwf=Pwf1+Pwf2Pwfr1+Pwfr2 Region II: Pwfr1<Pwf=Pwf1+Pwf2Pwfr1+Pwfr2 Region I: 0<Pwf=Pwf1+Pwf2Pwfr1+Pwfr1 Pwfr1 Pwfr2

work page 2004