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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