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AECOM

aecom

The Secret Hidden Power of Questions

የጥያቄዎች ምስጢር ድብቅ ኃይል

القوة الخفية للأسئلة

El poder secreto oculto de las preguntas

O poder secreto oculto das preguntas

Հարցերի գաղտնի թաքնված ուժը

質問の秘密の隠された力

კითხვების საიდუმლო ფარული ძალა

Nguya ya Sekele oyo ebombami ya mituna

د پوښتنو پټ ځواک

Tapukuypa Pakasqa Pakasqa Atiynin

Қувваи махфии саволҳо

问题隐藏的秘密力量

Table of Links Acknowledgements 1 Introduction to thesis 1.1 History and Evidence 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.5 Summary Appendices A Standard model B Friedmann equations C Type I seasaw mechanism D Feynman diagrams in two-component DM model Bibliography 4.5 Summary 5 Summary This thesis examines several models of Weakly Interacting Massive Particle (WIMP) dark matter and their implications from a phenomenological standpoint. These models are particularly intriguing due to the stringent constraints of direct detection experiments. In chapters 2 and 4, we explore models where the pseudo-scalar properties serve as both a mediator and a dark matter candidate, allowing them to evade the direct detection bounds. Additionally, in chapter 3, we delve into a two-component dark matter model and its relevant phenomenological features. Here, we provide a concise overview of each of these chapters. Chapter 2 focuses on an ALP portal fermion dark matter model. This model is minimal, consisting of three right-handed neutrinos (RHNs) and one pseudo-scalar mediator known as ALP. The two RHNs mix with the three active neutrinos, generating neutrino masses through Type I seesaw mechanism (detailed in Appendix-C). The stability of the third RHN is ensured by a Z2 symmetry, making it a viable dark matter candidate. We then proceed to analyze the constraints imposed by neutrino mixing and ALP couplings to Standard Model (SM) particles. By incorporating these experimental bounds, we investigate the parameter space that allows for fermion dark matter, taking into account relic density, direct detection, and indirect detection constraints. Remarkably, the model exhibits a significant parameter space that satisfies the relic density constraint, and it also possesses detectable signatures for indirect searches such as Fermi-LAT or HESS. In chapter 3, we explore a model of dark matter consisting of a scalar and fermion as components. This model is based on a generic U(1)𝑋 extension of the Standard Model (SM). To ensure the consistency of the model, we introduce three right-handed neutrinos, which necessitate the presence of the U(1)𝑋 gauge symmetry. Additionally, we incorporate two new complex scalars into the model. These scalars are singlets under the SM but carry charges under the U(1)𝑋 gauge symmetry. One of the scalars breaks the U(1)𝑋 gauge symmetry, while the other scalar is responsible for the scalar dark matter (DM). In this model, we also include two Z2 symmetries. One of these symmetries applies to the right-handed neutrinos, while the other symmetry applies to the new scalar. As a result, both the scalars and fermions can potentially act as DM candidates. Furthermore, the two remaining right-handed neutrinos mix with the three active neutrinos, leading to the generation of neutrino masses through a Type I seesaw mechanism Finally, we explore a few more U(1) models that allow for parameter space accommodating the two-component DM. This model offers a rich phenomenology and provides opportunities for testing and collaboration with colleagues. We have previously discussed the WIMP DM models within the framework of the ALP portal and U(1)𝑋 extension. These models have been thoroughly analyzed, taking into account various theoretical and experimental constraints. The model we have discussed addresses certain issues associated with WIMP models, particularly the lack of direct detection of WIMP DM. Additionally, our models incorporate neutrino physics, including oscillation and mass mechanisms. These bottom-up models offer a fascinating range of phenomenology that can be explored through collider-based experiments. However, it is worth noting that the standard DM mass in these WIMP DM models is limited to the range of a few GeV to TeV, which restricts the scope of our explored DM scenarios. There are other intriguing mass ranges for DM, such as ultralight dark matter, sub-GeV DM, and primordial black holes, among others. I intend to investigate these scenarios in future research. Furthermore, apart from considering different DM candidates, exploring the production mechanisms of DM is also an intriguing avenue for further exploration and study. This paper is available on arxiv under CC BY 4.0 DEED license. Author:
(1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Table of Links Acknowledgements Acknowledgements 1 Introduction to thesis 1 Introduction to thesis 1.1 History and Evidence 1.1 History and Evidence 1.2 Facts on dark matter 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.4 Dark matter detection 1.5 Outline of the thesis 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.2 Model 2.3 Existing constraints on ALP parameter space 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.4 Dark matter analysis 2.5 Summary 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.2 Model 3.3 Theoretical and experimental constraints 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.2 Model 4.3 Theoretical and experimental constraints 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.4 Dark Matter analysis 4.5 Summary 4.5 Summary Appendices Appendices A Standard model A Standard model B Friedmann equations B Friedmann equations C Type I seasaw mechanism C Type I seasaw mechanism D Feynman diagrams in two-component DM model D Feynman diagrams in two-component DM model Bibliography Bibliography 4.5 Summary 5 Summary This thesis examines several models of Weakly Interacting Massive Particle (WIMP) dark matter and their implications from a phenomenological standpoint. These models are particularly intriguing due to the stringent constraints of direct detection experiments. In chapters 2 and 4, we explore models where the pseudo-scalar properties serve as both a mediator and a dark matter candidate, allowing them to evade the direct detection bounds. Additionally, in chapter 3, we delve into a two-component dark matter model and its relevant phenomenological features. Here, we provide a concise overview of each of these chapters. Chapter 2 focuses on an ALP portal fermion dark matter model. This model is minimal, consisting of three right-handed neutrinos (RHNs) and one pseudo-scalar mediator known as ALP. The two RHNs mix with the three active neutrinos, generating neutrino masses through Type I seesaw mechanism (detailed in Appendix-C). The stability of the third RHN is ensured by a Z2 symmetry, making it a viable dark matter candidate. We then proceed to analyze the constraints imposed by neutrino mixing and ALP couplings to Standard Model (SM) particles. By incorporating these experimental bounds, we investigate the parameter space that allows for fermion dark matter, taking into account relic density, direct detection, and indirect detection constraints. Remarkably, the model exhibits a significant parameter space that satisfies the relic density constraint, and it also possesses detectable signatures for indirect searches such as Fermi-LAT or HESS. In chapter 3, we explore a model of dark matter consisting of a scalar and fermion as components. This model is based on a generic U(1)𝑋 extension of the Standard Model (SM). To ensure the consistency of the model, we introduce three right-handed neutrinos, which necessitate the presence of the U(1)𝑋 gauge symmetry. Additionally, we incorporate two new complex scalars into the model. These scalars are singlets under the SM but carry charges under the U(1)𝑋 gauge symmetry. One of the scalars breaks the U(1)𝑋 gauge symmetry, while the other scalar is responsible for the scalar dark matter (DM). In this model, we also include two Z2 symmetries. One of these symmetries applies to the right-handed neutrinos, while the other symmetry applies to the new scalar. As a result, both the scalars and fermions can potentially act as DM candidates. Furthermore, the two remaining right-handed neutrinos mix with the three active neutrinos, leading to the generation of neutrino masses through a Type I seesaw mechanism Finally, we explore a few more U(1) models that allow for parameter space accommodating the two-component DM. This model offers a rich phenomenology and provides opportunities for testing and collaboration with colleagues. We have previously discussed the WIMP DM models within the framework of the ALP portal and U(1)𝑋 extension. These models have been thoroughly analyzed, taking into account various theoretical and experimental constraints. The model we have discussed addresses certain issues associated with WIMP models, particularly the lack of direct detection of WIMP DM. Additionally, our models incorporate neutrino physics, including oscillation and mass mechanisms. These bottom-up models offer a fascinating range of phenomenology that can be explored through collider-based experiments. However, it is worth noting that the standard DM mass in these WIMP DM models is limited to the range of a few GeV to TeV, which restricts the scope of our explored DM scenarios. There are other intriguing mass ranges for DM, such as ultralight dark matter, sub-GeV DM, and primordial black holes, among others. I intend to investigate these scenarios in future research. Furthermore, apart from considering different DM candidates, exploring the production mechanisms of DM is also an intriguing avenue for further exploration and study. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Author: Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai.

Dark Matter Study: Examining Several Models of WIMP

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How to Optimize Ethereum Gas Efficiency With Multidimensional EIP-1559

Jak optimalizovat účinnost etherea pomocí multidimenzionálního EIP-1559

Cómo optimizar la eficiencia del gas de Ethereum con EIP-1559 multidimensional

چگونه کارایی گاز اتریوم را با EIP-1559 چند بعدی بهینه کنیم

Comment optimiser l'efficacité du gaz Ethereum avec l'EIP-1559 multidimensionnel

Como optimizar a eficiencia do gas Ethereum con EIP-1559 multidimensional

כיצד לייעל את יעילות גז Ethereum עם EIP-1559 רב ממדי

Come ottimizzare l'efficienza del gas di Ethereum con EIP-1559 multidimensionale

多次元 EIP-1559 でイーサリアムのガス効率を最適化する方法

Kā optimizēt Ethereum gāzes efektivitāti, izmantojot daudzdimensiju EIP-1559

Како да се оптимизира ефикасноста на гасот Ethereum со повеќедимензионален EIP-1559

Олон хэмжээст EIP-1559 ашиглан Ethereum хийн үр ашгийг хэрхэн оновчтой болгох вэ

Ako optimalizovať účinnosť etherea pomocou multidimenzionálneho EIP-1559

Today, February 17, 2025, marks a significant milestone for the Pacific Islands as we officially launch the Development Phase of the Digital Moana Initiative. This groundbreaking project aims to enhance geospatial data management across the Pacific region by leveraging decentralized physical infrastructure networks (DePIN) and blockchain technology. The initiative will democratize access to high-resolution LiDAR and point cloud datasets, enabling more efficient resource management, disaster response, and economic opportunities for the region. The Case for DePIN in the Pacific Context The DePIN model has already proven its worth in various sectors, from Helium's wireless networks to Hivemapper's crowdsourced mapping solutions. The Pacific Islands present a unique opportunity to implement these decentralized principles in a region where data sharing and infrastructure coordination have historically been challenging. The Digital Moana initiative is now entering its Development Phase, where we will build a unified, blockchain-powered geospatial data ecosystem to efficiently manage and distribute high-resolution LiDAR data and dense point cloud datasets. Architectural Framework: A Regional Approach The Digital Moana infrastructure will establish a pilot network connecting six major Pacific nations: Papua New Guinea, Solomon Islands, Vanuatu, Fiji, Samoa, and Tonga, with plans to expand to the rest of the Pacific on a demand basis. Each node in this network will maintain its local infrastructure, including PostgreSQL/PostGIS databases and processing capabilities, while remaining interconnected through a cryptographic distributed cloud portal on IPFS (InterPlanetary File System). This system is specifically designed to avoid a single point-of-failure in handling the complexities of large-scale LiDAR and point cloud data management, enabling efficient storage, processing, and distribution of these detailed datasets. At this point in time we are considering Aleph Cloud as a potential platform for implementing this decentralized geospatial infrastructure. During the Development Phase, the Digital Moana initiative will focus on the systems development of its core infrastructure, emphasizing the database, data upload mechanisms, and full-stack back-end and front-end systems. The database development will involve designing and implementing a distributed PostgreSQL/PostGIS architecture, with back-end nodes deployed across participating Pacific Islands. This decentralized approach ensures that each island maintains local control over its data while contributing to a unified regional network. For data upload systems, we will develop secure and scalable pipelines to facilitate seamless integration of data from aerial and drone surveys, ground control points, and historical datasets into the distributed database. The front-end will be hosted on Aleph Cloud with IPFS, providing a decentralized and resilient user interface for data visualization, subscription management, and analytics. This setup ensures that users can access the platform reliably, even in low-bandwidth environments, while maintaining the decentralized ethos of the initiative. Together, these components will form a cohesive, full-stack system designed to support the Digital Moana initiative's goals of decentralized data management and accessibility. Blockchain-Powered Data Access Design A key feature of the Digital Moana initiative is its innovative use of NFT-based subscription management for LiDAR and point cloud datasets. During the Development Phase, we will implement a transparent and efficient system where data access rights are tokenized. Each dataset subscription will be represented by a unique NFT containing crucial metadata about geographic coverage, point cloud density, classification levels, resolution, and access permissions. This approach will address several longstanding challenges in geospatial data distribution: Transparent ownership and clear audit trails of data usage
Automated compliance through smart contracts
Enhanced analytics capabilities for resource optimization
Real-time tracking of data utilization
Efficient distribution of large-scale point cloud datasets
Decentralized data management and access
No single point of failure Quality Control and Local Empowerment Framework The Development Phase will also incorporate a robust quality control framework to ensure data integrity while respecting local autonomy. Each participating nation will maintain its processing capabilities while benefiting from decentralized quality assessment protocols. This includes specialized workflows for LiDAR data validation, ensuring accuracy in point cloud classification, and maintaining consistent data quality standards across the region. This hybrid approach will allow for standardized data quality while accommodating local needs and requirements. Synchronization System Development The Development Phase will include the creation of a sophisticated synchronization service to maintain data consistency across all nodes, with specific consideration for the unique challenges of managing large LiDAR datasets. This system will ensure that updates and new datasets are efficiently distributed while respecting bandwidth limitations common in Pacific Island nations. The decentralized cloud portal will act as an orchestrator, managing data flow and access rights while maintaining the decentralized nature of the network and implementing efficient compression and streaming protocols for point cloud data. Economic Implications and Sustainability Considerations By implementing an e-commerce system integrated with blockchain technology, the Digital Moana platform will create new economic opportunities for Pacific Island nations. The platform will allow for the monetization of high-value LiDAR and point cloud datasets while ensuring fair compensation for data providers. This economic model will provide a sustainable funding source for ongoing data collection and infrastructure maintenance. Security and Future Resilience Planning The Development Phase will prioritize security, incorporating multi-factor authentication, encrypted data delivery, distributed web presence, and regular smart contract audits. These measures will ensure that sensitive geospatial data remains protected while maintaining accessibility for authorized users. Future enhancements could include secondary markets for data subscriptions, advanced analytics dashboards for point cloud visualization and analysis, and automated feature extraction capabilities. Implementation Vision The Digital Moana initiative represents more than just a technological upgrade; it's a paradigm shift in how Pacific Island nations manage and share their geospatial resources. By embracing DePIN principles and blockchain technology, this system will create a more equitable, efficient, and sustainable approach to geospatial data management, particularly for high-resolution LiDAR and point cloud datasets. As we embark on the Development Phase, we envision a future of increased regional cooperation, improved disaster response capabilities, and new economic opportunities emerging from this digital transformation. The success of Digital Moana could serve as a model for other regions seeking to implement decentralized infrastructure solutions for geospatial data management. Let's Go! Today, February 17, 2025, marks a significant milestone for the Pacific Islands as we officially launch the Development Phase of the Digital Moana Initiative . This groundbreaking project aims to enhance geospatial data management across the Pacific region by leveraging decentralized physical infrastructure networks (DePIN) and blockchain technology. The initiative will democratize access to high-resolution LiDAR and point cloud datasets, enabling more efficient resource management, disaster response, and economic opportunities for the region. Development Phase Digital Moana Initiative The Case for DePIN in the Pacific Context The DePIN model has already proven its worth in various sectors, from Helium 's wireless networks to Hivemapper 's crowdsourced mapping solutions. The Pacific Islands present a unique opportunity to implement these decentralized principles in a region where data sharing and infrastructure coordination have historically been challenging. The Digital Moana initiative is now entering its Development Phase, where we will build a unified, blockchain-powered geospatial data ecosystem to efficiently manage and distribute high-resolution LiDAR data and dense point cloud datasets. Helium Hivemapper Architectural Framework: A Regional Approach The Digital Moana infrastructure will establish a pilot network connecting six major Pacific nations: Papua New Guinea, Solomon Islands, Vanuatu, Fiji, Samoa, and Tonga, with plans to expand to the rest of the Pacific on a demand basis. Each node in this network will maintain its local infrastructure, including PostgreSQL/PostGIS databases and processing capabilities, while remaining interconnected through a cryptographic distributed cloud portal on IPFS (InterPlanetary File System). This system is specifically designed to avoid a single point-of-failure in handling the complexities of large-scale LiDAR and point cloud data management, enabling efficient storage, processing, and distribution of these detailed datasets. At this point in time we are considering Aleph Cloud as a potential platform for implementing this decentralized geospatial infrastructure. Aleph Cloud During the Development Phase, the Digital Moana initiative will focus on the systems development of its core infrastructure, emphasizing the database, data upload mechanisms, and full-stack back-end and front-end systems. The database development will involve designing and implementing a distributed PostgreSQL/PostGIS architecture, with back-end nodes deployed across participating Pacific Islands. This decentralized approach ensures that each island maintains local control over its data while contributing to a unified regional network. For data upload systems, we will develop secure and scalable pipelines to facilitate seamless integration of data from aerial and drone surveys, ground control points, and historical datasets into the distributed database. The front-end will be hosted on Aleph Cloud with IPFS, providing a decentralized and resilient user interface for data visualization, subscription management, and analytics. This setup ensures that users can access the platform reliably, even in low-bandwidth environments, while maintaining the decentralized ethos of the initiative. Together, these components will form a cohesive, full-stack system designed to support the Digital Moana initiative's goals of decentralized data management and accessibility. Blockchain-Powered Data Access Design A key feature of the Digital Moana initiative is its innovative use of NFT-based subscription management for LiDAR and point cloud datasets. During the Development Phase, we will implement a transparent and efficient system where data access rights are tokenized. Each dataset subscription will be represented by a unique NFT containing crucial metadata about geographic coverage, point cloud density, classification levels, resolution, and access permissions. This approach will address several longstanding challenges in geospatial data distribution: Transparent ownership and clear audit trails of data usage Automated compliance through smart contracts Enhanced analytics capabilities for resource optimization Real-time tracking of data utilization Efficient distribution of large-scale point cloud datasets Decentralized data management and access No single point of failure Transparent ownership and clear audit trails of data usage Automated compliance through smart contracts Enhanced analytics capabilities for resource optimization Real-time tracking of data utilization Efficient distribution of large-scale point cloud datasets Decentralized data management and access No single point of failure Quality Control and Local Empowerment Framework The Development Phase will also incorporate a robust quality control framework to ensure data integrity while respecting local autonomy. Each participating nation will maintain its processing capabilities while benefiting from decentralized quality assessment protocols. This includes specialized workflows for LiDAR data validation, ensuring accuracy in point cloud classification, and maintaining consistent data quality standards across the region. This hybrid approach will allow for standardized data quality while accommodating local needs and requirements. Synchronization System Development The Development Phase will include the creation of a sophisticated synchronization service to maintain data consistency across all nodes, with specific consideration for the unique challenges of managing large LiDAR datasets. This system will ensure that updates and new datasets are efficiently distributed while respecting bandwidth limitations common in Pacific Island nations. The decentralized cloud portal will act as an orchestrator, managing data flow and access rights while maintaining the decentralized nature of the network and implementing efficient compression and streaming protocols for point cloud data. Economic Implications and Sustainability Considerations By implementing an e-commerce system integrated with blockchain technology, the Digital Moana platform will create new economic opportunities for Pacific Island nations. The platform will allow for the monetization of high-value LiDAR and point cloud datasets while ensuring fair compensation for data providers. This economic model will provide a sustainable funding source for ongoing data collection and infrastructure maintenance. Security and Future Resilience Planning The Development Phase will prioritize security, incorporating multi-factor authentication, encrypted data delivery, distributed web presence, and regular smart contract audits. These measures will ensure that sensitive geospatial data remains protected while maintaining accessibility for authorized users. Future enhancements could include secondary markets for data subscriptions, advanced analytics dashboards for point cloud visualization and analysis, and automated feature extraction capabilities. Implementation Vision The Digital Moana initiative represents more than just a technological upgrade; it's a paradigm shift in how Pacific Island nations manage and share their geospatial resources. By embracing DePIN principles and blockchain technology, this system will create a more equitable, efficient, and sustainable approach to geospatial data management, particularly for high-resolution LiDAR and point cloud datasets. As we embark on the Development Phase, we envision a future of increased regional cooperation, improved disaster response capabilities, and new economic opportunities emerging from this digital transformation. The success of Digital Moana could serve as a model for other regions seeking to implement decentralized infrastructure solutions for geospatial data management. Let's Go!

Pacific Islands Break Ground on Decentralized Geospatial Infrastructure Development

Authors:
(1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China;
(4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China;
(5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China;
(6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 6. Open Questions 7. Conclusion Appendix A and References 4. Generalizability and Applicability Evaluations 4.1. Predictor and Response Generalizability While PPFL methods are often constrained in their choice of predictors and responses due to specific physical formulations, DPFL methods generally offer a more flexible framework. This flexibility indicates the potential for DPFL approaches to accommodate arbitrary known variables (including 𝑃 and 𝑄 of the PQ buses, 𝑉 of the slack and PV buses, and 𝜃 of the slack bus) as predictors, and arbitrary unknown variables (including 𝑃 of the slack bus, 𝑄 of the slack and PV buses, 𝑉 of the PQ buses, 𝜃 of the PQ and PV buses, and all active/reactive line flows, i.e., PF, PT, QF, QT) as responses. However, due to various factors, not all DPFL methods can achieve this level of generalizability: • For the PLS_BDL and PLS_BDLY2 methods, using 𝑉 , 𝑃 , and 𝑄 as predictors is enforced by the bundle strategy [6] they adopt. This strategy, designed to address variations in bus types, inherently constrains the choice of predictors by pre-defining known and unknown variables. • The LCP_BOX and LCP_JGD methods integrate physical knowledge of power flows by formulating constraints based on the Jacobian matrix derived from AC power flow equations expressed in polar coordinates [6]. Within this Jacobian matrix, 𝑃 and 𝑄 are treated as known variables, while 𝑉 and 𝜃 are considered unknown variables. • The LCP_COU method is specifically designed to linearly estimate the values of branch flows by leveraging the terminal voltages and angles as predictive variables [6]. Consequently, the method can only employ 𝑉 and 𝜃 as predictors, and treat PF, PT, QF, and QT as responses. • For the DC_LS and DLPF_C methods, since they incorporate the DC and DLPF models respectively into their framework [6], they must align their selection of predictors and responses with the underlying physical models they adopt. Note that the constrained flexibility in choosing predictors and responses results in notable limitations. First, these methods might not leverage all available known data for model training, leading to potential information loss. For instance, the DC_LS method only uses measurements of 𝑃 , disregarding a large amount of known voltage data. Second, the capability to predict unknown variables using the developed linear model may be restricted. For example, the LCP_BOX method is restricted to calculating branch flow values, leading to a quite limited functional scope. 4.2. Applicability to Cases with Multicollinearity The ordinary least squares method struggles with multicollinearity [6]. Methods LS, LS_CLS, and LS_REC share this limitation, since they are all based on the ordinary least squares framework. Additionally, LS_SVD and LS_TOL are also affected by this problem, as discussed in [6]. Subsequent experiments will numerically demonstrate their limitations in this context. 4.3. Zero Predictor Applicability The issue of zero predictors arises when certain known variable measurements in the training dataset are consistently zero. A typical example is the inclusion of the slack bus angle in the predictor set, whose value is commonly set as zero and remains zero throughout. Other instances may involve PQ buses where active/reactive power consumption is zero during the measurement period. This situation leads to zero columns in the predictor dataset matrix (where columns represent different variables, and rows represent individual measurements). Not all DPFL methods can handle these zero columns effectively: • Methods based on ordinary least squares, including LS, LS_CLS, and LS_REC, have difficulties with zero predictors, as these zero columns render the Gram matrix of the predictor matrix non-invertible, thereby leading to the failure of these methods. 4.4. Constant Predictor Applicability The constant predictor issue extends beyond the zero predictor problem, occurring when measurements of certain variables in the training dataset remain constant, not necessarily zero. A typical example is the fixed terminal voltages at PV buses, which normally stay constant over the measurement period, resulting in constant-value columns in the predictor matrix. 4.5. Normalization Applicability As detailed in [6], incorporating physical knowledge into DPFL methods can become problematic with datasets normalized via variance-scaling techniques, like unit-energy normalization, where each variable is normalized independently. This independent normalization disrupts the inherent physical relationships among variables, such as those represented in the Jacobian matrix or through coupling relationships, rendering methods like RR_VCS, LCP_BOX, LCP_COU, LCP_JGD, DC_LS, and DLPF_C inapplicable. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. Authors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Authors: Authors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.2. Evaluation Overview 5.3. Failure Evaluation 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 5.5. Efficiency Evaluation 6. Open Questions 6. Open Questions 7. Conclusion 7. Conclusion Appendix A and References Appendix A and References 4. Generalizability and Applicability Evaluations 4.1. Predictor and Response Generalizability While PPFL methods are often constrained in their choice of predictors and responses due to specific physical formulations, DPFL methods generally offer a more flexible framework. This flexibility indicates the potential for DPFL approaches to accommodate arbitrary known variables (including 𝑃 and 𝑄 of the PQ buses, 𝑉 of the slack and PV buses, and 𝜃 of the slack bus) as predictors, and arbitrary unknown variables (including 𝑃 of the slack bus, 𝑄 of the slack and PV buses, 𝑉 of the PQ buses, 𝜃 of the PQ and PV buses, and all active/reactive line flows, i.e., PF, PT, QF, QT) as responses. However, due to various factors, not all DPFL methods can achieve this level of generalizability: • For the PLS_BDL and PLS_BDLY2 methods, using 𝑉 , 𝑃 , and 𝑄 as predictors is enforced by the bundle strategy [6] they adopt. This strategy, designed to address variations in bus types, inherently constrains the choice of predictors by pre-defining known and unknown variables. • The LCP_BOX and LCP_JGD methods integrate physical knowledge of power flows by formulating constraints based on the Jacobian matrix derived from AC power flow equations expressed in polar coordinates [6]. Within this Jacobian matrix, 𝑃 and 𝑄 are treated as known variables, while 𝑉 and 𝜃 are considered unknown variables. • The LCP_COU method is specifically designed to linearly estimate the values of branch flows by leveraging the terminal voltages and angles as predictive variables [6]. Consequently, the method can only employ 𝑉 and 𝜃 as predictors, and treat PF, PT, QF, and QT as responses. • For the DC_LS and DLPF_C methods, since they incorporate the DC and DLPF models respectively into their framework [6], they must align their selection of predictors and responses with the underlying physical models they adopt. Note that the constrained flexibility in choosing predictors and responses results in notable limitations. First, these methods might not leverage all available known data for model training, leading to potential information loss. For instance, the DC_LS method only uses measurements of 𝑃 , disregarding a large amount of known voltage data. Second, the capability to predict unknown variables using the developed linear model may be restricted. For example, the LCP_BOX method is restricted to calculating branch flow values, leading to a quite limited functional scope. 4.2. Applicability to Cases with Multicollinearity The ordinary least squares method struggles with multicollinearity [6]. Methods LS, LS_CLS, and LS_REC share this limitation, since they are all based on the ordinary least squares framework. Additionally, LS_SVD and LS_TOL are also affected by this problem, as discussed in [6]. Subsequent experiments will numerically demonstrate their limitations in this context. 4.3. Zero Predictor Applicability The issue of zero predictors arises when certain known variable measurements in the training dataset are consistently zero. A typical example is the inclusion of the slack bus angle in the predictor set, whose value is commonly set as zero and remains zero throughout. Other instances may involve PQ buses where active/reactive power consumption is zero during the measurement period. This situation leads to zero columns in the predictor dataset matrix (where columns represent different variables, and rows represent individual measurements). Not all DPFL methods can handle these zero columns effectively: • Methods based on ordinary least squares, including LS, LS_CLS, and LS_REC, have difficulties with zero predictors, as these zero columns render the Gram matrix of the predictor matrix non-invertible, thereby leading to the failure of these methods. 4.4. Constant Predictor Applicability The constant predictor issue extends beyond the zero predictor problem, occurring when measurements of certain variables in the training dataset remain constant, not necessarily zero. A typical example is the fixed terminal voltages at PV buses, which normally stay constant over the measurement period, resulting in constant-value columns in the predictor matrix. 4.5. Normalization Applicability As detailed in [6], incorporating physical knowledge into DPFL methods can become problematic with datasets normalized via variance-scaling techniques, like unit-energy normalization, where each variable is normalized independently. This independent normalization disrupts the inherent physical relationships among variables, such as those represented in the Jacobian matrix or through coupling relationships, rendering methods like RR_VCS, LCP_BOX, LCP_COU, LCP_JGD, DC_LS, and DLPF_C inapplicable. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. available on arxiv

When Power Flow Models Go Off the Rails

Authors:
(1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China;
(4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China;
(5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China;
(6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 6. Open Questions 7. Conclusion Appendix A and References 3. Review of Existing Experiments Before conducting evaluations of the methods listed in Table 1, we here aim to offer a detailed review of existing DPFL experiments in the literature, as depicted in Table 2. This review intends to present the experimental accomplishments of previous DPFL studies, while simultaneously revealing the importance and need for an extensive numerical comparison of all DPFL methods. Below are the further discussions of Table 2. Firstly, Table 2 indicates that both transmission and distribution grids were used as test cases to verify DPFL methods. While distribution grids differ from transmission grids in terms of symmetry and topology, DPFL methods are generally applicable to both types of systems, such as the methods in [15, 12, 30, 21, 11]. The reasons are twofold. First, even if the three phases in distribution systems are unbalanced, DPFL methods can still be implemented by either training one DPFL model for each phase [52], or training an overall DPFL model for all variables in three phases [22, 11, 12, 24]. Note that the latter can generate a DPFL model reflecting the mutual influences between phases. Second, from a DPFL perspective, radial topologies do not pose any unique challenges compared to mesh topologies, as the difference is only in the number of dependent and independent variables. In summary, while distribution grids may have unbalanced characteristics and radial topologies, these attributes do not bring special difficulties to DPFL studies. Secondly, Table 2 reveals that many evaluations solely depended on artificial data for training and testing, without considering the effects of noise and outliers on the data. This ideal testing environment is rarely found in real life, and the resulting conclusions may not hold in practice. To address this issue, it is recommended that synthetic data be injected with noise and outliers to mimic real-world scenarios. Thirdly, as indicated in Table 2, only a few studies report the load fluctuation range used in their simulations. This is worth mentioning because the accuracy of the DPFL model is highly dependent on the simulated fluctuations. E.g., a narrow fluctuation range typically leads to higher accuracy for the DPFL model. Without this information being made public, it is difficult to determine the reason for the high accuracy of the evaluated DPFL model. Finally, Table 2 indicates which of the DPFL approaches have been evaluated in existing DPFL studies. These evaluations aimed to implement a comparative analysis between established and novel DPFL methods at that time. However, the scope of these comparisons is quite narrow, with only a few DPFL studies undertaking evaluations against a limited number of existing DPFL approaches (some works only conducted comparisons with PPFL methods). Such constrained comparisons fail to provide an in-depth understanding of the overall performance of DPFL methods. Consequently, a more exhaustive and inclusive comparison across all the DPFL approaches is clearly needed, in order to demonstrate their relative merits and limitations comprehensively. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. Authors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Authors: Authors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.2. Evaluation Overview 5.3. Failure Evaluation 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 5.5. Efficiency Evaluation 6. Open Questions 6. Open Questions 7. Conclusion 7. Conclusion Appendix A and References Appendix A and References 3. Review of Existing Experiments Before conducting evaluations of the methods listed in Table 1, we here aim to offer a detailed review of existing DPFL experiments in the literature, as depicted in Table 2. This review intends to present the experimental accomplishments of previous DPFL studies, while simultaneously revealing the importance and need for an extensive numerical comparison of all DPFL methods. Below are the further discussions of Table 2. Firstly, Table 2 indicates that both transmission and distribution grids were used as test cases to verify DPFL methods. While distribution grids differ from transmission grids in terms of symmetry and topology, DPFL methods are generally applicable to both types of systems, such as the methods in [15, 12, 30, 21, 11]. The reasons are twofold. First, even if the three phases in distribution systems are unbalanced, DPFL methods can still be implemented by either training one DPFL model for each phase [52], or training an overall DPFL model for all variables in three phases [22, 11, 12, 24]. Note that the latter can generate a DPFL model reflecting the mutual influences between phases. Second, from a DPFL perspective, radial topologies do not pose any unique challenges compared to mesh topologies, as the difference is only in the number of dependent and independent variables. In summary, while distribution grids may have unbalanced characteristics and radial topologies, these attributes do not bring special difficulties to DPFL studies. Secondly, Table 2 reveals that many evaluations solely depended on artificial data for training and testing, without considering the effects of noise and outliers on the data. This ideal testing environment is rarely found in real life, and the resulting conclusions may not hold in practice. To address this issue, it is recommended that synthetic data be injected with noise and outliers to mimic real-world scenarios. Thirdly, as indicated in Table 2, only a few studies report the load fluctuation range used in their simulations. This is worth mentioning because the accuracy of the DPFL model is highly dependent on the simulated fluctuations. E.g., a narrow fluctuation range typically leads to higher accuracy for the DPFL model. Without this information being made public, it is difficult to determine the reason for the high accuracy of the evaluated DPFL model. Finally, Table 2 indicates which of the DPFL approaches have been evaluated in existing DPFL studies. These evaluations aimed to implement a comparative analysis between established and novel DPFL methods at that time. However, the scope of these comparisons is quite narrow, with only a few DPFL studies undertaking evaluations against a limited number of existing DPFL approaches (some works only conducted comparisons with PPFL methods). Such constrained comparisons fail to provide an in-depth understanding of the overall performance of DPFL methods. Consequently, a more exhaustive and inclusive comparison across all the DPFL approaches is clearly needed, in order to demonstrate their relative merits and limitations comprehensively. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. available on arxiv

Why Current Power Flow Models May Not Work in Real-World Scenarios

ors:
(1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland;
(3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China;
(4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China;
(5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China;
(6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 6. Open Questions 7. Conclusion Appendix A and References 2. Evaluated Methods Table 1 enumerates the 44 evaluated methods, detailing for each the corresponding abbreviation, the training algorithm employed, and any supporting techniques utilized. The following points warrant attention. Firstly, for the linearly constrained programming approaches, we also evaluate these methods without their key constraints, e.g., the bound constraints, coupling constraints, or structure constraints, in order to verify the added value of incorporating such constraints. It is crucial to recognize that, even without these key constraints, the resulting programming models remain different, attributed to the varied supporting techniques they incorporate. Secondly, the first part of this tutorial [6] reveals the modular nature of DPFL studies, highlighting their flexibility in the assembly of various techniques to forge novel methodologies. In alignment with this paradigm, we introduce several methods previously unexplored within the DPFL domain, and include them into the following comparative analysis. These approaches include the least squares with pseudoinverse, least squares augmented by principal component analysis, generalized least squares with pseudoinverse, and a clustering-based version of the partial least squares[1]. It is crucial to clarify that the objective of integrating these methods is not to argue their “novelty/superiority” over all pre-existing techniques. Rather, we intend to demonstrate the ease with which one can deviate from conventional paths to devise distinct methodologies. Notably, some of these introduced methods have demonstrated satisfying performance and rankings in subsequent evaluations. This outcome, particularly given the unsophisticated-designed nature of these approaches, suggests a high potential for further advancements in DPFL research. Finally, our evaluation also encloses a selection of physics-driven power flow linearization (PPFL) methods, such as the classic DC model, the power transfer distribution factor model, the warm-start 1st-order Taylor approximation model (derived from the equations of nodal power injections in polar coordinates), and the decoupled linearized power flow model [7]. Note that these PPFL methods are widely recognized and employed in both academic research and industry practices. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. [1] In the adaptation of the least squares method to incorporate pseudoinverse, the conventional inversion operation used in the ordinary least squares method is substituted with the Moore–Penrose inverse. This adjustment is designed to enhance the method’s resilience to multicollinearity issues. Similarly, for the generalized least squares method augmented with pseudoinverse, the initial iteration of the well-known feasible generalized least squares method is modified to employ the least squares with pseudoinverse instead of the ordinary least squares. This modification also aims to strengthen the method’s ability to manage multicollinearity. In the case of clustering-based partial least squares, the approach involves substituting the ordinary least squares component within the clustering-based least squares methodology (as discussed in Part I [6]) with the ordinary partial least squares. This change seeks to better accommodate the inherent nonlinear characteristics of AC power flows. For details on the least squares with principal component analysis, the reader is referred to Appendix A. ors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. ors: ors: (1) Mengshuo Jia, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (2) Gabriela Hug, Department of Information Technology and Electrical Engineering, ETH Zürich, Physikstrasse 3, 8092, Zürich, Switzerland; (3) Ning Zhang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China; (4) Zhaojian Wang, Department of Automation, Shanghai Jiao Tong University, Dongchuan Rd 800, 200240, Shanghai, China; (5) Yi Wang, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong, China; (6) Chongqing Kang, Department of Electrical Engineering, Tsinghua University, Shuangqing Rd 30, 100084, Beijing, China. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Evaluated Methods 3. Review of Existing Experiments 3. Review of Existing Experiments 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4. Generalizability and Applicability Evaluations and 4.1. Predictor and Response Generalizability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.2. Applicability to Cases with Multicollinearity and 4.3. Zero Predictor Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 4.4. Constant Predictor Applicability and 4.5. Normalization Applicability 5. Numerical Evaluations and 5.1. Experiment Settings 5. Numerical Evaluations and 5.1. Experiment Settings 5.2. Evaluation Overview 5.2. Evaluation Overview 5.3. Failure Evaluation 5.3. Failure Evaluation 5.4. Accuracy Evaluation 5.4. Accuracy Evaluation 5.5. Efficiency Evaluation 5.5. Efficiency Evaluation 6. Open Questions 6. Open Questions 7. Conclusion 7. Conclusion Appendix A and References Appendix A and References 2. Evaluated Methods Table 1 enumerates the 44 evaluated methods, detailing for each the corresponding abbreviation, the training algorithm employed, and any supporting techniques utilized. The following points warrant attention. Firstly, for the linearly constrained programming approaches, we also evaluate these methods without their key constraints, e.g., the bound constraints, coupling constraints, or structure constraints, in order to verify the added value of incorporating such constraints. It is crucial to recognize that, even without these key constraints, the resulting programming models remain different, attributed to the varied supporting techniques they incorporate. Secondly, the first part of this tutorial [6] reveals the modular nature of DPFL studies, highlighting their flexibility in the assembly of various techniques to forge novel methodologies. In alignment with this paradigm, we introduce several methods previously unexplored within the DPFL domain, and include them into the following comparative analysis. These approaches include the least squares with pseudoinverse, least squares augmented by principal component analysis, generalized least squares with pseudoinverse, and a clustering-based version of the partial least squares[1]. It is crucial to clarify that the objective of integrating these methods is not to argue their “novelty/superiority” over all pre-existing techniques. Rather, we intend to demonstrate the ease with which one can deviate from conventional paths to devise distinct methodologies. Notably, some of these introduced methods have demonstrated satisfying performance and rankings in subsequent evaluations. This outcome, particularly given the unsophisticated-designed nature of these approaches, suggests a high potential for further advancements in DPFL research. Finally, our evaluation also encloses a selection of physics-driven power flow linearization (PPFL) methods, such as the classic DC model, the power transfer distribution factor model, the warm-start 1st-order Taylor approximation model (derived from the equations of nodal power injections in polar coordinates), and the decoupled linearized power flow model [7]. Note that these PPFL methods are widely recognized and employed in both academic research and industry practices. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. available on arxiv [1] In the adaptation of the least squares method to incorporate pseudoinverse, the conventional inversion operation used in the ordinary least squares method is substituted with the Moore–Penrose inverse. This adjustment is designed to enhance the method’s resilience to multicollinearity issues. Similarly, for the generalized least squares method augmented with pseudoinverse, the initial iteration of the well-known feasible generalized least squares method is modified to employ the least squares with pseudoinverse instead of the ordinary least squares. This modification also aims to strengthen the method’s ability to manage multicollinearity. In the case of clustering-based partial least squares, the approach involves substituting the ordinary least squares component within the clustering-based least squares methodology (as discussed in Part I [6]) with the ordinary partial least squares. This change seeks to better accommodate the inherent nonlinear characteristics of AC power flows. For details on the least squares with principal component analysis, the reader is referred to Appendix A.

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