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

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