The Trials and Triumphs of DPFL Research

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 7. Conclusion This paper, as the second part of a tutorial series, presents a comprehensive review of existing DPFL experiments, detailing the capabilities and limitations of these experiments. Additionally, it provides an in-depth numerical examination of DPFL methods, complementing the theoretical insights explored in Part I and addressing the limited experimental work found in the existing literature. This dual focus aims to offer a more holistic understanding of DPFL by bridging the gap between theory and practice. Specifically, this paper analyzes all the approaches’ generalizability towards the selection of predictor and response, as well as their applicability towards multicollinearity, zero predictor, constant predictor, and normalization. Then, through rigorous numerical simulations involving 44 different methods across numerous test cases scaling from 9-bus to 1354-bus, this paper has illustrated the practical performance that mere theoretical analyses could not reveal. The practical performance is evaluated from two angles: accuracy and computational efficiency. For the accuracy evaluation, this paper (i) clarifies the reasons behind the failures encountered by DPFL methods, (ii) provides insights into how DPFL methods compare with traditional PPFL approaches, (iii) offers comprehensive discussions on the uniform outcomes observed across various test cases for all DPFL methods, and (iv) dives into extensive analyses of individual DPFL methods, highlighting their distinctive performances. Regarding the evaluation of computational efficiency, the paper (i) explores the comparative efficiency of DPFL versus PPFL methods, (ii) offers in-depth discussions on the consistent performance metrics of DPFL methods across multiple test scenarios, and (iii) presents detailed evaluations of specific groups of DPFL methods, focusing on their similar performance. Furthermore, drawing on insights from this study and Part I [6], as well as the identified open questions, this paper outlines ten promising yet challenging directions, which sketch a roadmap for future research in DPFL. Addressing these questions will not only advance our understanding of DPFL methods but also contribute to the broader goal of developing more reliable, efficient, and accessible tools for power systems research, education, and applications. As for our next step, by leveraging our comprehensive repository of documentation and codebase for DPFL methods, we will start with the tenth suggested direction: creating an exhaustive, elegant, and user-friendly toolbox. This toolbox will fully support all DPFL methods and the related functionalities, aiming to offer significant ease of use for researchers and engineers, to support them in exploring and/or leveraging the area of DPFL, thereby reshaping the frontiers of the DPFL research and application. Appendix A References [1] Y. Chen, F. Pan, F. Qiu, A. S. Xavier, T. Zheng, M. Marwali, B. Knueven, Y. Guan, P. B. Luh, L. Wu, B. Yan, M. A. Bragin, H. Zhong, A. Giacomoni, R. Baldick, B. Gisin, Q. Gu, R. Philbrick, F. Li, Security-constrained unit commitment for electricity market: Modeling, solution methods, and future challenges, IEEE Transactions on Power Systems 38 (5) (2023) 4668–4681. doi:10.1109/TPWRS.2022. 3213001. [2] D. K. Molzahn, I. A. Hiskens, et al., A survey of relaxations and approximations of the power flow equations, Foundations and Trends® in Electric Energy Systems 4 (1-2) (2019) 1–221. [3] X. Li, K. Hedman, Data driven linearized ac power flow model with regression analysis, arXiv preprint arXiv:1811.09727 (2018). [4] Z. Li, J. Yu, Q. Wu, Approximate linear power flow using logarithmic transform of voltage magnitudes with reactive power and transmission loss consideration, IEEE Transactions on Power Systems 33 (4) (2017) 4593–4603. [5] M. Jia, G. Hug, Overview of data-driven power flow linearization, in: 2023 IEEE Belgrade PowerTech, 2023, pp. 01–06. doi:10.1109/ PowerTech55446.2023.10202779. [6] M. Jia, G. Hug, N. Zhang, Z. Wang, Y. Wang, C. Kang, Data-driven power flow linearization: Theory (2024). [7] J. Yang, N. Zhang, C. Kang, Q. Xia, A state-independent linear power flow model with accurate estimation of voltage magnitude, IEEE Transactions on Power Systems 32 (5) (2016) 3607–3617. [8] H. Xu, A. D. Domínguez-García, V. V. Veeravalli, P. W. Sauer, Datadriven voltage regulation in radial power distribution systems, IEEE Transactions on Power Systems 35 (3) (2019) 2133–2143. [9] Z. Shao, Q. Zhai, J. Wu, X. Guan, Data based linear power flow model: Investigation of a least-squares based approximation, IEEE Transactions on Power Systems 36 (5) (2021) 4246–4258. [10] S. hentong, Z. Qiaozhu, W. Jiang, G. Xiaohong, Data based linearization: Least-squares based approximation, arXiv preprint arXiv:2007.02494 (2020). [11] Y. Liu, Z. Li, Y. Zhou, Data-driven-aided linear three-phase power flow model for distribution power systems, IEEE Transactions on Power Systems (2021). [12] Y. Liu, Z. Li, Y. Zhou, A physics-based and data-driven linear threephase power flow model for distribution power systems, arXiv preprint arXiv:2103.10147 (2021). [13] Y. Liu, Y. Wang, N. Zhang, D. Lu, C. Kang, A data-driven approach to linearize power flow equations considering measurement noise, IEEE Transactions on Smart Grid 11 (3) (2019) 2576–2587. [14] S. Powell, A. Ivanova, D. Chassin, Fast solutions in power system simulation through coupling with data-driven power flow models for voltage estimation, arXiv preprint arXiv:2001.01714 (2020). [15] L. Guo, Y. Zhang, X. Li, Z. Wang, Y. Liu, L. Bai, C. Wang, Data-driven power flow calculation method: A lifting dimension linear regression approach, IEEE Transactions on Power Systems (2021). [16] M. Korda, I. Mezić, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica 93 (2018) 149–160. [17] Y. Liu, Z. Li, S. Sun, A data-driven method for online constructing linear power flow model, IEEE Transactions on Industry Applications (2023). [18] Y. Tan, Y. Chen, Y. Li, Y. Cao, Linearizing power flow model: A hybrid physical model-driven and data-driven approach, IEEE Transactions on Power Systems 35 (3) (2020) 2475–2478. [19] Y. Liu, N. Zhang, Y. Wang, J. Yang, C. Kang, Data-driven power flow linearization: A regression approach, IEEE Transactions on Smart Grid 10 (3) (2018) 2569–2580. [20] S. Nowak, Y. C. Chen, L. Wang, Measurement-based optimal der dispatch with a recursively estimated sensitivity model, IEEE Transactions on Power Systems 35 (6) (2020) 4792–4802. [21] Y. Chen, C. Wu, J. Qi, Data-driven power flow method based on exact linear regression equations, Journal of Modern Power Systems and Clean Energy (2021). [22] J. Chen, W. Wu, L. A. Roald, Data-driven piecewise linearization for distribution three-phase stochastic power flow, IEEE Transactions on Smart Grid (2021). [23] J. Zhang, Z. Wang, X. Zheng, L. Guan, C. Chung, Locally weighted ridge regression for power system online sensitivity identification considering data collinearity, IEEE Transactions on Power Systems 33 (2) (2017) 1624–1634. [24] J. Chen, W. Li, W. Wu, T. Zhu, Z. Wang, C. Zhao, Robust data-driven linearization for distribution three-phase power flow, in: 2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), IEEE, 2020, pp. 1527–1532. [25] Z. Shao, Q. Zhai, Z. Han, X. Guan, A linear ac unit commitment formulation: An application of data-driven linear power flow model, International Journal of Electrical Power & Energy Systems 145 (2023) 108673. [26] J. Yu, Y. Weng, R. Rajagopal, Robust mapping rule estimation for power flow analysis in distribution grids, in: 2017 North American Power Symposium (NAPS), IEEE, 2017, pp. 1–6. [27] J. Yu, Y. Weng, R. Rajagopal, Mapping rule estimation for power flow analysis in distribution grids, arXiv preprint arXiv:1702.07948 (2017). [28] P. Li, W. Wu, X. Wan, B. Xu, A data-driven linear optimal power flow model for distribution networks, IEEE Transactions on Power Systems (2022). [29] Y. Liu, B. Xu, A. Botterud, N. Zhang, C. Kang, Bounding regression errors in data-driven power grid steady-state models, IEEE Transactions on Power Systems 36 (2) (2020) 1023–1033. [30] Y. Liu, Z. Li, J. Zhao, Robust data-driven linear power flow model with probability constrained worst-case errors, arXiv preprint arXiv:2112.10320 (2021). [31] X. Li, Fast heuristic ac power flow analysis with data-driven enhanced linearized model, Energies 13 (13) (2020) 3308. [32] M. Jia, Q. Cao, C. Shen, G. Hug, Frequency-control-aware probabilistic load flow: An analytical method, IEEE Transactions on Power Systems (2022) 1–16doi:10.1109/TPWRS.2022.3223884. [33] H. Wold, Path models with latent variables: The nipals approach, in: Quantitative sociology, Elsevier, 1975, pp. 307–357. [34] Ieee pes test feeders (2018). URL https://cmte.ieee.org/pes-testfeeders/ [35] Commercial and residential hourly load profiles for all tmy3 locations in the united states (2022). URL https://data.openei.org/submissions/153 [36] I. Pena, C. B. Martinez-Anido, B.-M. Hodge, An extended ieee 118-bus test system with high renewable penetration, IEEE Transactions on Power Systems 33 (1) (2017) 281–289. [37] S. Bolognani, Approximate linear solution of power flow equations in power distribution networks (2015). URL http://github.com/saveriob/approx-pf [38] A. B. Birchfield, T. Xu, K. M. Gegner, K. S. Shetye, T. J. Overbye, Grid structural characteristics as validation criteria for synthetic networks, IEEE Transactions on power systems 32 (4) (2016) 3258–3265. [39] R. D. Zimmerman, C. E. Murillo-Sánchez, Matpower 6.0 user’s manual, Power Systems Engineering Research Center 9 (2016). [40] M. E. Baran, F. F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Power Engineering Review 9 (4) (1989) 101–102. [41] S. Bolognani, S. Zampieri, On the existence and linear approximation of the power flow solution in power distribution networks, IEEE Transactions on Power Systems 31 (1) (2015) 163–172. [42] Y. Wang, N. Zhang, H. Li, J. Yang, C. Kang, Linear three-phase power flow for unbalanced active distribution networks with pv nodes, CSEE Journal of Power and Energy Systems 3 (3) (2017) 321–324. [43] W. Zheng, W. Wu, B. Zhang, C. Lin, Distributed optimal residential demand response considering operational constraints of unbalanced distribution networks, IET generation, transmission & distribution 12 (9) (2018) 1970–1979. [44] J. Chen, W. Wu, L. A. Roald, Supplemental file for data-driven piecewise linearization for distribution three-phase stochastic power flow (2021). URL https://www.dropbox.com/s/wowq9xwsn6tbd4h/Supplemental%20File.docx?dl=0 [45] J. Zhang, X. Zheng, Z. Wang, L. Guan, C. Chung, Power system sensitivity identification—inherent system properties and data quality, IEEE Transactions on Power Systems 32 (4) (2016) 2756–2766. [46] Y. Fu, M. Shahidehpour, Z. Li, Security-constrained unit commitment with ac constraints, IEEE transactions on power systems 20 (2) (2005) 1001–1013. [47] Data published by electrical and computer engineering department at illinois institute of technology. URL http://motor.ece.iit.edu/Data/ [48] G. E. Constante-Flores, A. J. Conejo, F. Qiu, Ac network-constrained unit commitment via relaxation and decomposition, IEEE Transactions on Power Systems 37 (3) (2022) 2187–2196. doi:10.1109/TPWRS.2021. 3120180. [49] X. Li, P. Balasubramanian, M. Abdi-Khorsand, A. S. Korad, K. W. Hedman, Effect of topology control on system reliability: Tva test case, in: CIGRE US National Committee Grid of the Future Symposium, 2014. [50] A. B. Birchfield, T. Xu, K. M. Gegner, K. S. Shetye, T. J. Overbye, Grid structural characteristics as validation criteria for synthetic networks, IEEE Transactions on Power Systems 32 (4) (2017) 3258–3265. doi:10.1109/TPWRS.2016.2616385. [51] C. Draxl, B. Hodge, A. Clifton, J. McCaa, Overview and meteorological validation of the wind integration national dataset toolkit, Tech. rep., National Renewable Energy Lab.(NREL), Golden, CO (United States) (2015). [52] C. Mugnier, K. Christakou, J. Jaton, M. De Vivo, M. Carpita, M. Paolone, Model-less/measurement-based computation of voltage sensitivities in unbalanced electrical distribution networks, in: 2016 Power Systems Computation Conference (PSCC), IEEE, 2016, pp. 1–7. [53] M. Jia, G. Hug, N. Zhang, Z. Wang, Y. Wang, C. Kang, Open-source code for data-driven power flow linearization - part ii: Simulation (the code will be public once the paper is accepted) (2024). [54] M. Brown, M. Biswal, S. Brahma, S. J. Ranade, H. Cao, Characterizing and quantifying noise in pmu data, in: 2016 IEEE Power and Energy Society General Meeting (PESGM), IEEE, 2016, pp. 1–5. [55] M. Jia, G. Hug, N. Zhang, Z. Wang, Y. Wang, C. Kang, Data-driven power flow linearization - supplementary file (2024). URL https://drive.google.com/file/d/ 1jf7rB8iQS5IJZsq7fDwTl08ueBS2s1wk/view?usp=drive_link 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 7. Conclusion This paper, as the second part of a tutorial series, presents a comprehensive review of existing DPFL experiments, detailing the capabilities and limitations of these experiments. Additionally, it provides an in-depth numerical examination of DPFL methods, complementing the theoretical insights explored in Part I and addressing the limited experimental work found in the existing literature. This dual focus aims to offer a more holistic understanding of DPFL by bridging the gap between theory and practice. Specifically, this paper analyzes all the approaches’ generalizability towards the selection of predictor and response, as well as their applicability towards multicollinearity, zero predictor, constant predictor, and normalization. Then, through rigorous numerical simulations involving 44 different methods across numerous test cases scaling from 9-bus to 1354-bus, this paper has illustrated the practical performance that mere theoretical analyses could not reveal. The practical performance is evaluated from two angles: accuracy and computational efficiency. For the accuracy evaluation, this paper (i) clarifies the reasons behind the failures encountered by DPFL methods, (ii) provides insights into how DPFL methods compare with traditional PPFL approaches, (iii) offers comprehensive discussions on the uniform outcomes observed across various test cases for all DPFL methods, and (iv) dives into extensive analyses of individual DPFL methods, highlighting their distinctive performances. Regarding the evaluation of computational efficiency, the paper (i) explores the comparative efficiency of DPFL versus PPFL methods, (ii) offers in-depth discussions on the consistent performance metrics of DPFL methods across multiple test scenarios, and (iii) presents detailed evaluations of specific groups of DPFL methods, focusing on their similar performance. Furthermore, drawing on insights from this study and Part I [6], as well as the identified open questions, this paper outlines ten promising yet challenging directions, which sketch a roadmap for future research in DPFL. Addressing these questions will not only advance our understanding of DPFL methods but also contribute to the broader goal of developing more reliable, efficient, and accessible tools for power systems research, education, and applications. As for our next step, by leveraging our comprehensive repository of documentation and codebase for DPFL methods, we will start with the tenth suggested direction: creating an exhaustive, elegant, and user-friendly toolbox. This toolbox will fully support all DPFL methods and the related functionalities, aiming to offer significant ease of use for researchers and engineers, to support them in exploring and/or leveraging the area of DPFL, thereby reshaping the frontiers of the DPFL research and application. Appendix A References [1] Y. Chen, F. Pan, F. Qiu, A. S. Xavier, T. Zheng, M. Marwali, B. Knueven, Y. Guan, P. B. Luh, L. Wu, B. Yan, M. A. Bragin, H. Zhong, A. Giacomoni, R. Baldick, B. Gisin, Q. Gu, R. Philbrick, F. Li, Security-constrained unit commitment for electricity market: Modeling, solution methods, and future challenges, IEEE Transactions on Power Systems 38 (5) (2023) 4668–4681. doi:10.1109/TPWRS.2022. 3213001. [2] D. K. Molzahn, I. A. Hiskens, et al., A survey of relaxations and approximations of the power flow equations, Foundations and Trends® in Electric Energy Systems 4 (1-2) (2019) 1–221. [3] X. Li, K. Hedman, Data driven linearized ac power flow model with regression analysis, arXiv preprint arXiv:1811.09727 (2018). [4] Z. Li, J. Yu, Q. Wu, Approximate linear power flow using logarithmic transform of voltage magnitudes with reactive power and transmission loss consideration, IEEE Transactions on Power Systems 33 (4) (2017) 4593–4603. [5] M. Jia, G. Hug, Overview of data-driven power flow linearization, in: 2023 IEEE Belgrade PowerTech, 2023, pp. 01–06. doi:10.1109/ PowerTech55446.2023.10202779. [6] M. Jia, G. Hug, N. Zhang, Z. Wang, Y. Wang, C. Kang, Data-driven power flow linearization: Theory (2024). [7] J. Yang, N. Zhang, C. Kang, Q. Xia, A state-independent linear power flow model with accurate estimation of voltage magnitude, IEEE Transactions on Power Systems 32 (5) (2016) 3607–3617. [8] H. Xu, A. D. Domínguez-García, V. V. Veeravalli, P. W. Sauer, Datadriven voltage regulation in radial power distribution systems, IEEE Transactions on Power Systems 35 (3) (2019) 2133–2143. [9] Z. Shao, Q. Zhai, J. Wu, X. Guan, Data based linear power flow model: Investigation of a least-squares based approximation, IEEE Transactions on Power Systems 36 (5) (2021) 4246–4258. [10] S. hentong, Z. Qiaozhu, W. Jiang, G. Xiaohong, Data based linearization: Least-squares based approximation, arXiv preprint arXiv:2007.02494 (2020). [11] Y. Liu, Z. Li, Y. Zhou, Data-driven-aided linear three-phase power flow model for distribution power systems, IEEE Transactions on Power Systems (2021). [12] Y. Liu, Z. Li, Y. Zhou, A physics-based and data-driven linear threephase power flow model for distribution power systems, arXiv preprint arXiv:2103.10147 (2021). [13] Y. Liu, Y. Wang, N. Zhang, D. Lu, C. Kang, A data-driven approach to linearize power flow equations considering measurement noise, IEEE Transactions on Smart Grid 11 (3) (2019) 2576–2587. [14] S. Powell, A. Ivanova, D. Chassin, Fast solutions in power system simulation through coupling with data-driven power flow models for voltage estimation, arXiv preprint arXiv:2001.01714 (2020). [15] L. Guo, Y. Zhang, X. Li, Z. Wang, Y. Liu, L. Bai, C. Wang, Data-driven power flow calculation method: A lifting dimension linear regression approach, IEEE Transactions on Power Systems (2021). [16] M. Korda, I. Mezić, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica 93 (2018) 149–160. [17] Y. Liu, Z. Li, S. Sun, A data-driven method for online constructing linear power flow model, IEEE Transactions on Industry Applications (2023). [18] Y. Tan, Y. Chen, Y. Li, Y. Cao, Linearizing power flow model: A hybrid physical model-driven and data-driven approach, IEEE Transactions on Power Systems 35 (3) (2020) 2475–2478. [19] Y. Liu, N. Zhang, Y. Wang, J. Yang, C. Kang, Data-driven power flow linearization: A regression approach, IEEE Transactions on Smart Grid 10 (3) (2018) 2569–2580. [20] S. Nowak, Y. C. Chen, L. Wang, Measurement-based optimal der dispatch with a recursively estimated sensitivity model, IEEE Transactions on Power Systems 35 (6) (2020) 4792–4802. [21] Y. Chen, C. Wu, J. Qi, Data-driven power flow method based on exact linear regression equations, Journal of Modern Power Systems and Clean Energy (2021). [22] J. Chen, W. Wu, L. A. Roald, Data-driven piecewise linearization for distribution three-phase stochastic power flow, IEEE Transactions on Smart Grid (2021). [23] J. Zhang, Z. Wang, X. Zheng, L. Guan, C. Chung, Locally weighted ridge regression for power system online sensitivity identification considering data collinearity, IEEE Transactions on Power Systems 33 (2) (2017) 1624–1634. [24] J. Chen, W. Li, W. Wu, T. Zhu, Z. Wang, C. Zhao, Robust data-driven linearization for distribution three-phase power flow, in: 2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), IEEE, 2020, pp. 1527–1532. [25] Z. Shao, Q. Zhai, Z. Han, X. Guan, A linear ac unit commitment formulation: An application of data-driven linear power flow model, International Journal of Electrical Power & Energy Systems 145 (2023) 108673. [26] J. Yu, Y. Weng, R. Rajagopal, Robust mapping rule estimation for power flow analysis in distribution grids, in: 2017 North American Power Symposium (NAPS), IEEE, 2017, pp. 1–6. [27] J. Yu, Y. Weng, R. Rajagopal, Mapping rule estimation for power flow analysis in distribution grids, arXiv preprint arXiv:1702.07948 (2017). [28] P. Li, W. Wu, X. Wan, B. Xu, A data-driven linear optimal power flow model for distribution networks, IEEE Transactions on Power Systems (2022). [29] Y. Liu, B. Xu, A. Botterud, N. Zhang, C. Kang, Bounding regression errors in data-driven power grid steady-state models, IEEE Transactions on Power Systems 36 (2) (2020) 1023–1033. [30] Y. Liu, Z. Li, J. Zhao, Robust data-driven linear power flow model with probability constrained worst-case errors, arXiv preprint arXiv:2112.10320 (2021). [31] X. Li, Fast heuristic ac power flow analysis with data-driven enhanced linearized model, Energies 13 (13) (2020) 3308. [32] M. Jia, Q. Cao, C. Shen, G. Hug, Frequency-control-aware probabilistic load flow: An analytical method, IEEE Transactions on Power Systems (2022) 1–16doi:10.1109/TPWRS.2022.3223884. [33] H. Wold, Path models with latent variables: The nipals approach, in: Quantitative sociology, Elsevier, 1975, pp. 307–357. [34] Ieee pes test feeders (2018). URL https://cmte.ieee.org/pes-testfeeders/ https://cmte.ieee.org/pes-testfeeders/ [35] Commercial and residential hourly load profiles for all tmy3 locations in the united states (2022). URL https://data.openei.org/submissions/153 https://data.openei.org/submissions/153 [36] I. Pena, C. B. Martinez-Anido, B.-M. Hodge, An extended ieee 118-bus test system with high renewable penetration, IEEE Transactions on Power Systems 33 (1) (2017) 281–289. [37] S. Bolognani, Approximate linear solution of power flow equations in power distribution networks (2015). URL http://github.com/saveriob/approx-pf http://github.com/saveriob/approx-pf [38] A. B. Birchfield, T. Xu, K. M. Gegner, K. S. Shetye, T. J. Overbye, Grid structural characteristics as validation criteria for synthetic networks, IEEE Transactions on power systems 32 (4) (2016) 3258–3265. [39] R. D. Zimmerman, C. E. Murillo-Sánchez, Matpower 6.0 user’s manual, Power Systems Engineering Research Center 9 (2016). [40] M. E. Baran, F. F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Power Engineering Review 9 (4) (1989) 101–102. [41] S. Bolognani, S. Zampieri, On the existence and linear approximation of the power flow solution in power distribution networks, IEEE Transactions on Power Systems 31 (1) (2015) 163–172. [42] Y. Wang, N. Zhang, H. Li, J. Yang, C. Kang, Linear three-phase power flow for unbalanced active distribution networks with pv nodes, CSEE Journal of Power and Energy Systems 3 (3) (2017) 321–324. [43] W. Zheng, W. Wu, B. Zhang, C. 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This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. available on arxiv