Inference of dominant modes for linear stochastic processes

For dynamical systems that can be modelled as asymptotically stable linear systems forced by Gaussian noise, this paper develops methods to infer (estimate) their dominant modes from observations in real time. The modes can be real or complex. For a real mode (monotone decay), the goal is to infer its damping rate and mode shape. For a complex mode (oscillatory decay), the goal is to infer its frequency, damping rate and (complex) mode shape. Their amplitudes and correlations are encoded in a mode covariance matrix that is also to be inferred. The work is motivated and illustrated by the problem of detection of oscillations in power flow in AC electrical networks. Suggestions of some other applications are given.

The main comment about the paper is that the methodology has not been properly validated, i.e. we do not really know whether it works or not. For testing his method, the author used data presented in Fig. 4. There are two problems with that data. The first is that Fig. 4 presents a frequency trace with 1 Hz time resolution while PMU measurements are taken with the frequency of about 30-60 Hz. Resolution of 1 Hz is not high enough for the estimation of power system dynamics. Secondly, the data is from a single measurement point so it is not possible to check the author's claimed benefit of being able to process measurements from many PMUs simultaneously.
The author claims that he could not use National Grid data due to confidentiality reasons but it is unclear why he did not use Fig. 1 which presumably shows actual PMU data and has a much higher time resolution than Fig. 4. Also, Fig. 4 shows data from 8 PMUs so the author could check his claim that his method could process data from many PMUs simultaneously.
Generally, the problem of not being able to access real data is quite common and a standard approach is then to simulate it. This is quite simple and would require simulating a simple power system using standard software (e.g. Matpower, PSAT, PST or any other), add noise to the signals, estimate the dominant modes and compare them to the actual ones (known as we know the underlying model). The advantage of using simulated data is that we know the actual model so we can check how accurate the estimates are.
The second comment is editorial. The paper presents a general, textbook-like treatment of the problem rather than a derivation of a methodology. The treatment is quite wide (46 pages), aiming to be complete, with a sprawling narrative and many diversions. The reader is distracted from the main message and has to go through a lot of supplementary material which is often only marginally related to, or used for, the proposed methodology (for example, sections 3 and 4 -Langevin process, OU process). I am not sure if such a comprehensive treatment is acceptable (or perhaps even desired) for this journal but I personally would prefer a more disciplined and focused treatment concentrating on the derivation of the methodology and relegating a lot of background and supplementary material to appendices.
To summarise, the paper presents an interesting and potentially useful piece of mathematics but its practical usefulness is still to be proved. In my personal opinion, the presentation could be made more focused but I would defer to the Editor and the author to decide whether or not to follow this recommendation.

Review form: Reviewer 3
Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation? Accept as is
Comments to the Author(s) Thanks, I enjoyed your introductory review and the application to a power grid and possible other applications.

Review form: Reviewer 4
Is the manuscript scientifically sound in its present form? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
Motivated by National Grid's need to detect nascent oscillations in power flow fast enough to activate controller strategies, the author considers and develops a method of detecting oscillations in a fluctuating signal in real time. He does this through treatment of a model dynamical system subject to Gaussian noise. I found it difficult to get an intuitive feeling for the extent to which the approach has succeeded in solving the original problem. For example, looking at Fig 1, the oscillations become evident by eye at about 608 or 609 s, though a little earlier on one of the green traces. So how much sooner than this can the new method reliably identify growth of the incipient oscillations? This is an interesting, scholarly, and highly erudite manuscript. So far as I can tell, it is correct and original and meets all of the RSOS criteria, so I am happy to recommend that it be accepted for publication. Have you any concerns about statistical analyses in this paper? No

Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
The author has written an interesting paper on a technique that could have potential within power systems. The paper is quite long with much general information before we come to the core method. The background information is useful but such a relatively detailed treatment could have been moved to an appendix. The paper is written in a bit personal style and I presume that this is acceptable for a uni-personal paper but possible there should be no reference to family members and co-workers. The paper will benefit from a small example demonstrating the approach. I would like to compliment the author for clear statements of the deficiencies of the current implementation. Maybe statements like this need to be removed: (we might extend this unpublished paper to include a report on some tests).

Decision letter (RSOS-201442.R0)
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Dear Dr MacKay
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Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). 1. There is no clear problem formulation and analysis in the manuscript. I guess that this manuscript just review some issues on Gaussian processes and stochastic problems.
2-The structure of the introduction and whole manuscript are disordered and needs to be reorganized.
3-The results and simulations are not clear to verify the derived achievements. The results must be structured in the form of theorems, remarks, ... .

Reviewer: 2
Comments to the Author(s) First, a necessary disclosure. I am a power engineer, not a mathematician, so my interest is from the application point of view. Hence, I am not able to assess the presented mathematics and my assessment is concentrated on whether or not the presented methodology could be of interest from power system application point of view. Generally, the answer to that question is yes (providing that the methodology works, about which later). The author claims two main advantages of the presented methodology. Firstly, it utilises Kalman filter to process information as it comes along. Secondly, it processes information from many measurement points (Phasor Measurement Units, PMUs) simultaneously. This compares favourably with currently used methodologies which typically operate on a moving time window (i.e. operate in a batch mode) and process information from one PMU at a time. The latter means that it may be difficult to identify the same mode at different locations as its estimated frequency and damping may have slightly different values in different locations.
The main comment about the paper is that the methodology has not been properly validated, i.e. we do not really know whether it works or not. For testing his method, the author used data presented in Fig. 4. There are two problems with that data. The first is that Fig. 4 presents a frequency trace with 1 Hz time resolution while PMU measurements are taken with the frequency of about 30-60 Hz. Resolution of 1 Hz is not high enough for the estimation of power system dynamics. Secondly, the data is from a single measurement point so it is not possible to check the author's claimed benefit of being able to process measurements from many PMUs simultaneously.
The author claims that he could not use National Grid data due to confidentiality reasons but it is unclear why he did not use Fig. 1 which presumably shows actual PMU data and has a much higher time resolution than Fig. 4. Also, Fig. 4 shows data from 8 PMUs so the author could check his claim that his method could process data from many PMUs simultaneously.
Generally, the problem of not being able to access real data is quite common and a standard approach is then to simulate it. This is quite simple and would require simulating a simple power system using standard software (e.g. Matpower, PSAT, PST or any other), add noise to the signals, estimate the dominant modes and compare them to the actual ones (known as we know the underlying model). The advantage of using simulated data is that we know the actual model so we can check how accurate the estimates are.
The second comment is editorial. The paper presents a general, textbook-like treatment of the problem rather than a derivation of a methodology. The treatment is quite wide (46 pages), aiming to be complete, with a sprawling narrative and many diversions. The reader is distracted from the main message and has to go through a lot of supplementary material which is often only marginally related to, or used for, the proposed methodology (for example, sections 3 and 4 -Langevin process, OU process). I am not sure if such a comprehensive treatment is acceptable (or perhaps even desired) for this journal but I personally would prefer a more disciplined and focused treatment concentrating on the derivation of the methodology and relegating a lot of background and supplementary material to appendices.
To summarise, the paper presents an interesting and potentially useful piece of mathematics but its practical usefulness is still to be proved. In my personal opinion, the presentation could be made more focused but I would defer to the Editor and the author to decide whether or not to follow this recommendation.
Reviewer: 3 Comments to the Author(s) Thanks, I enjoyed your introductory review and the application to a power grid and possible other applications. This is an interesting, scholarly, and highly erudite manuscript. So far as I can tell, it is correct and original and meets all of the RSOS criteria, so I am happy to recommend that it be accepted for publication.

------------------------------------A couple of typos:
Page/line Comment 3/17 "processes" 32/34 "data are" [data are plural -there are also other instances] Reviewer: 5 Comments to the Author(s) The author has written an interesting paper on a technique that could have potential within power systems. The paper is quite long with much general information before we come to the core method. The background information is useful but such a relatively detailed treatment could have been moved to an appendix. The paper is written in a bit personal style and I presume that this is acceptable for a uni-personal paper but possible there should be no reference to family members and co-workers. The paper will benefit from a small example demonstrating the approach. I would like to compliment the author for clear statements of the deficiencies of the current implementation. Maybe statements like this need to be removed: (we might extend this unpublished paper to include a report on some tests).

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See Appendix A.
Decision letter (RSOS-201442.R1) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr MacKay, It is a pleasure to accept your manuscript entitled "Inference of dominant modes for linear stochastic processes" in its current form for publication in Royal Society Open Science.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience@royalsociety.org) and the production office (openscience_proofs@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal.
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The results must be structured in the form of theorems, remarks, ... . I feel this is not appropriate for this paper. It would give a misleading impression that it is all about rigorous formulation and proofs. The results are variants on known results and are