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The voltage follower , as shown in Figure 1, uses virtual short and virtual break. At first glance, it seems simple and clear. If you think that there is not much content to pay attention to, then you may be wrong. Understanding the op amp's voltage-following circuit is a great help for understanding the op amp's in-phase, inverting, differential, and various op amp circuits. circuit of the op amp Figure 1 op amp voltage follower circuit Voltage follower circuit analysis If we connect the output of the op amp to its inverting input and then apply a voltage signal to the noninverting input, we will find that the output voltage of the op amp will follow the input voltage very well. Assume that the input and output voltages of the initial state op amp are both , and then when Vin increases from , will also increase in the direction of positive voltage. This is because it is assumed that Vin suddenly increases, still has no response and is still . is greater than 0, so multiply the open-loop gain of the op amp. , so that the output of the op amp begins to increase in the direction of the positive voltage. 0V 0V Vout Vout 0V Ve=Vin-Vout Vout=Ve*A Vout When increases, the output voltage is fed back to the inverting input, which then reduces the voltage difference between the two inputs of the op amp, ie, will decrease, in the same open-loop gain case. Next, will naturally decrease. Vout Ve Vout The end result is that no matter how large the input voltage is (of course, within the op amp's input voltage range), the op amp will always output a voltage that is very close to , but this output voltage is just below . There is a sufficient voltage difference between the two inputs of the guaranteed op amp to maintain the output of the op amp, which is . Vin Vout Vin Ve Vout=Ve*A Negative feedback in op amp circuits This circuit will soon reach a steady state, the amplitude of the output voltage will accurately maintain the voltage difference between the two input terminals of the op amp, which in turn will produce an accurate amplitude of the op amp output voltage. Connecting the output of the op amp to the inverting input of the op amp is called negative feedback, which is the key to self-stabilizing the system. This applies not only to op amps, but also to any common dynamic system. This stability allows the op amp to have the ability to operate in a linear mode, rather than just being saturated, fully "on" or "off", just as it is used for comparators without any negative feedback. Due to the high gain of the op amp, the voltage maintained at the inverting input of the op amp is almost equal to . For example, an op amp has an open loop gain of 200,000. If is equal to 6V, then the output voltage will be 5.999 970 000 149 999V. This produces a sufficient voltage difference at the input of the op amp, =6V-5.999 970 000 149 999V=29.999 85uV, which is amplified and then produces a voltage of 5.999 970 000 149 999V at the output, thus the system Will be stable here. Vin Vin Ve As you can see, 29.999 85uV is a small voltage, so for practical calculations, we can think that the voltage difference between the two inputs of the op amp maintained by negative feedback is . The whole process is shown in Figure 2. This is also the “virtual short” we are familiar with, and since the impedance between the two inputs of the op amp is very large, there is naturally a “virtual break”. The circuit below has a stable double-loop closed-loop gain, and the output voltage simply follows the input voltage. Ve=0V Figure 2 The role of negative feedback One big advantage of using negative feedback is that we don't have to worry about the actual voltage gain of the op amp, as long as it is large enough. If the op amp's voltage gain is not 200 0000 but 250 000, this will cause the op amp's output voltage to be closer to , and the voltage difference between the smaller inputs will be used to produce the desired output voltage. In the circuit illustrated in Figure 2, the output voltage is also equal to the input voltage at the inverting input of the op amp. Therefore, for the circuit design engineer, in order to achieve a stable closed-loop gain of the amplifier circuit, the open-loop gain of the op amp does not have to be an accurate value, and the negative feedback causes the system to self-adjust. Vin Using negative feedback improves linearity, gain stability, output impedance, and gain accuracy, but using negative feedback also poses a serious problem, which is to reduce system stability for voltage-following circuits with unity gain. This is the worst case scenario, especially when driving capacitive loads, interested students can check the relevant information themselves. Regarding the op amp circuit, many times we are infused with the inverting end to follow the in-phase end. As mentioned above, can't we follow the inverting end of the phase? For the voltage follower circuit mentioned today, only the inverting terminal follows the non-inverting terminal. Here, if a positive input voltage is applied to the inverting terminal and the output is connected to the non-inverting terminal, and the output is assumed to be 0. will be a negative voltage, multiplied by the open-loop gain of the op amp, and the output will be a A negative voltage, returning to the non-inverting input of the op amp, will further result in a larger negative absolute voltage difference. Soon the output of the op amp will be saturated, and naturally the in-phase will not follow the inverting end. Ve However, for an op amp, if a reference voltage is applied to the inverting terminal, and other electronic components, such as a triode, MOS, etc., the overall loop of the op amp forms a negative feedback, and the in-phase end can also follow the inverting end. And this naturally breaks the rule that the inverting end of the familiar op amp follows the in-phase end. The voltage of the op amp follows the circuit, "virtual short", "virtual break" is the surface, and negative feedback is the root. Based on this root, it can help us understand the ever-changing op amp circuit.

The Voltage Follower Circuit of The Operational Amplifier

Since the reverse withstand voltage of the is limited, when used in a high- voltage environment, it is often necessary to use plurality of step-recovery rectifier diodes in series to meet the need of reverse withstand voltage. Due to the inconsistency in the volt-ampere characteristics, turn-on time, and recovery charges of the diode during the production process, the voltage sharing problem occurs when used in series, which causes the reverse voltage of a certain diode to be too high and damaged, disturbing the normal operation of other diodes, which ultimately affects the reliability and stability of the entire device. step-recovery diode 1. Characteristics and Factors of Uneven Voltage of the Diode 1.1 Characteristics Diodes are very commonly-used power electronic devices. Generally, we are easier to miss the key parameters of familiar devices, and only care about macroscopic parameters on the macro level, such as reverse withstand voltage, on-state current, and reverse leakage current. While the junction capacitance, turn-off and turn-on characteristics of the diode are easily overlooked. 1.2 Uneven Voltage Problem The main reason for the voltage sharing problem may be caused by either internal reasons or external ones. The internal reasons is related to the processing technology, and the external cause is about the external circuit. The volt-ampere characteristics of the diodes produced in the same batch are inconsistent, resulting in static uneven voltage of the diode; the inconsistency between the reverse recovery time and the on state state causes the dynamic uneven voltage of the diode. External circuit design can cause stray inductance and capacitance, which can cause uneven voltage problems in high-voltage high-frequency environments. 2. Analysis of the Uneven Voltage Problem 2.1 Analysis in Macroscopic Scale At the macro level, the diodes in series does not need to have voltage sharing, the main consideration is the influence of the diode itself. As shown in Figure 1, the two diodes are connected in series with an external reverse DC voltage. The diode with a small reverse saturation current can withstand a large voltage as the two diodes are connected in series, and the reverse saturation current is constant under an externally applied voltage rating. As shown in Figure 2, assumed that there is difference in the reverse saturation current between the two diodes, and the reverse saturation current of D2 is smaller, the above conclusion can be clearly drawn. FIG 1. Circuit of Two Diodes Connected in Series In actual operation, macroscopically, diodes have different voltage drops due to their own differences as shown in Figure 2. When the the voltage drop on D2 from the external voltage U increases reaches the critical point, the voltage drop of D1 is relatively small and still in the safe and stable region due to its large reverse saturation currents. FIG 2. Volt-ampere characteristics Diagram of Diode According to the above analysis, when U increases continuously. , the voltage drop on D2 will break through the critical transition voltage, causing the current to increase sharply. As D1 and D2 are connected in series in the main circuit, the current of D1 certainly increase with D2. However, the volt-ampere curve of D1 shows that the reverse voltage drop of D1 should reach the breakover voltage when a large current pass through it, so the sum of u1 and u2 is greater than U, which proves the speculation above is not true. Therefore, when U increases, the voltage of D2 will not continue to increase, and the voltage of D1 will keep increasing until its value is more than the sum of the reverse withstand voltages of the two tubes, at which point diode breakdown occurs. 2.2 Analysis in Microscopic Scale The pins and the layout of the diode on the board naturally evolve into and inductance in a high-voltage and high-frequency environment. The introduction of stray capacitance and inductance directly affects the turn-on and turn-off waveforms of the diode. stray capacitance The introduction of capacitance prevents a sudden change in voltage and the introduction of an  inductance prevents a sudden change in current. Figure 3 is the equivalent circuit diagram of the diodes in series at high frequency. In the figure, C1 is the junction capacitance, R is the reverse resistance, C2 is the stray capacitance at the high voltage, C3 is the stray capacitance formed by the diode to the ground. FIG 3. equivalent Circuit Diagram of series diodes at high frequency 3. Applications of Diodes in Series The high-frequency power supply of the electrostatic precipitator outputs high-frequency PWM waves, and the waves pass through the boosting transformer and the rectifier module to finally become an approximate linear voltage waveform. The rectifier module is integrated in the boosting transformer, and the diodes are in series. Because the voltage level is relatively high, it is generally considered to reach 10KV or more. Since the requirement for the accuracy of the output voltage waveform is not high, the diode can be directly connected in series, and the high-frequency rectifier diode is selected. The rectified output simulation waveform is shown in FIG. 4, and the measured waveform is shown in FIG. 5. FIG 4. Simulation Waveform FIG 5.  Measured Waveform As shown in the figures, the rectified output voltage has high and low peaks due to the influence of stray parameters. With the continuous development of science and technology, the manufacturing process of diodes is constantly improving, and the influence of its own factors is negligible. 4. Conclusion By simply analyzing the voltage sharing problem of the from both macroscopic and microscopic perspectives, it is concluded that when the diode is used in series in a high-voltage high-frequency environment, the diode itself will generate uneven voltage but not cause breakdown. series diode The uneven voltage generated by external factors is the result of breakdown of the diode, and it will cause even more serious problems.

Misunderstanding of Voltage Sharing Problem of Series Diode 

Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations 2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations 2.2. Globalization Strategies 2.2. Globalization Strategies 2.3. Sensitivity Analysis 2.3. Sensitivity Analysis 2.4. Matrix Coloring & Sparse Automatic Differentiation 2.4. Matrix Coloring & Sparse Automatic Differentiation 3. Special Capabilities 3. Special Capabilities 3.1. Composable Building Blocks 3.1. Composable Building Blocks 3.2. Smart PolyAlgortihm Defaults 3.2. Smart PolyAlgortihm Defaults 3.3. Non-Allocating Static Algorithms inside GPU Kernels 3.3. Non-Allocating Static Algorithms inside GPU Kernels 3.4. Automatic Sparsity Exploitation 3.4. Automatic Sparsity Exploitation 3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods 3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods 4. Results and 4.1. Robustness on 23 Test Problems 4. Results and 4.1. Robustness on 23 Test Problems 4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model 4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model 4.3. Large Ill-Conditioned Nonlinear Brusselator System 4.3. Large Ill-Conditioned Nonlinear Brusselator System 5. Conclusion and References 5. Conclusion and References 4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model Initializing DAEs is a crucial step in the numerical solution process, as it ensures the consistency and well-posedness of the problem. Improper initialization can lead to convergence issues, inaccurate solutions, or even complete failure of the numerical solver. The initialization of DAEs involves determining consistent initial values for the differential and algebraic variables that satisfy the algebraic constraints at the initial time. This process is essential for ensuring that the numerical solution starts from a physically meaningful state and avoids violating the system constraints from the outset. We generate a 32-dimensional DAE initialization problem for a Doyle-FullerNewman (DFN) battery model[15] [58, 59] from open-circuit voltage (OCV) charged at a high current [60, 59]. We benchmark the following methods: • TrustRegion from NonlinearSolve.jl, NLsolve.jl, and MINPACK. • NewtonRaphson from NonlinearSolve.jl, Sundials, and NLsolve.jl. • Quasi-Newton Methods from NonlinearSolve.jl. • LevenbergMarquardt from NonlinearSolve.jl and MINPACK. In Figure 9, we show that all the tested NonlinearSolve.jl solvers (except QuasiNewton Methods[16]) solve the initialization problem. Sundials and MINPACK both are unable to solve the problem. NLsolve.jl successfully solves the problem, but their performance is consistently worse. Finally, we note that our polyalgorithm (without ever being tuned to this problem) successfully solves this problem. 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 [15] https://help.juliahub.com/batteries/stable/ [16] Quasi-Newton Methods are known to fail on this problem [60]. Authors:
(1) AVIK PAL, CSAIL MIT, Cambridge, MA;
(2) FLEMMING HOLTORF;
(3) AXEL LARSSON;
(4) TORKEL LOMAN;
(5) UTKARSH;
(6) FRANK SCHÄFER;
(7) QINGYU QU;
(8) ALAN EDELMAN;
(9) CHRIS RACKAUCKAS, CSAIL MIT, Cambridge, MA. Authors: Authors: (1) AVIK PAL, CSAIL MIT, Cambridge, MA; (2) FLEMMING HOLTORF; (3) AXEL LARSSON; (4) TORKEL LOMAN; (5) UTKARSH; (6) FRANK SCHÄFER; (7) QINGYU QU; (8) ALAN EDELMAN; (9) CHRIS RACKAUCKAS, CSAIL MIT, Cambridge, MA.

Initializing the Doyle-Fuller-Newman (DFN) Battery Model

Turn Your Dumb Solar Inverter Into a Smart One With This Home Assistant Hack

With global electric vehicle sales expected to surpass those of conventional cars, EV battery remanufacturing could lead to cost savings and a smaller environmental impact. In 2025, electric vehicle sales are expected to grow by 29.9%, reaching 15.1 million units globally, up from 11.6 million in 2024. This will raise EVs' market share from 13.2% to 16.7%, according to S&P Global Mobility. The IEA forecasts that by 2030, electric cars will account for 65% of global sales. As the EV fleet grows, so will the volume of batteries reaching the end of their life. McKinsey predicts that by 2030, the U.S. will have 40 GWh of used batteries available for resale. That is why remanufacturing is crucial both for the environment and EV battery circularity. When EVs were first introduced, the industry assumed batteries wouldn't fail. As a result, many faulty batteries with healthy parts are being prematurely recycled. Remanufacturing eliminates the need for replacement batteries. Currently, when a battery malfunctions or reaches the end of its useful life, most vehicle owners replace the entire battery pack — often at an exorbitant cost that can exceed $10,000. Entrepreneurs have already spotted a promising opportunity. For example, Tesla cofounder JB Straubel founded Redwood Materials to create a "remanufacturing economy" by extracting and reusing raw materials from used batteries. The startup is on track to hit $200 million in revenue. The EV Battery Remanufacturing Process Unlike traditional manufacturing, remanufacturing is focused on giving old components a second life instead of starting from scratch. With battery packs, this means carefully restoring them to like-new condition. First, each pack undergoes a safety check to ensure it's safe to open, given the high voltage (400 to 800 volts). Next, a thorough inspection looks for any obvious issues, followed by advanced diagnostics with specialized tools. After diagnosing the issue, the battery is carefully disassembled, cleaned, and its components thoroughly inspected. In many cases, only certain parts, such as cells or modules, need replacing, while the rest—when feasible—can be reused. This approach significantly reduces the cost compared to purchasing a brand-new battery pack. Once the damaged parts are replaced with new or refurbished ones, the battery pack is reassembled and undergoes extensive testing to ensure it's safe and fully prepared for use in a new vehicle. This process results in remanufactured battery packs that perform like new. It’s also eco-friendly, reducing waste by reusing components instead of discarding entire packs. The many Benefits of Restoring a Single Battery By repairing and reusing battery packs, we lessen the environmental impact of battery disposal. This process helps to minimize the need for new raw materials, including rare minerals like lithium, cobalt, and nickel, which are often extracted in environmentally damaging ways. Remanufacturing also allows us to extend the life of existing batteries. Instead of manufacturing new batteries for every vehicle, we can keep using older packs by replacing only the damaged components, significantly increasing their operational lifespan. This is a win-win for both the environment and consumers, who benefit from lower replacement costs. When a battery is beyond repair, we can recycle it in a way that minimizes environmental damage. Most U.S. facilities use advanced machinery to safely and efficiently extract valuable materials. The materials we extract during the recycling process — such as lithium, cobalt, and nickel — can be sent back to battery manufacturers like LG and Panasonic. By investing in technologies that boost the lifespan of EV batteries, we can pave the way for a future where remanufacturing and recycling are key to a more sustainable transportation system. The future of transport is electric, and with EV remanufacturing, we can make it greener, more efficient, and more affordable for everyone. About the author Aminur Rahman is a leading expert in manufacturing across several industries, including automotive, electric vehicles (EVs), and biotech. He is known for promoting the circular economy and creating sustainable solutions that enhance business performance while benefiting the planet. Through his work, he is helping shape the future of technology toward a more eco-friendly and economically viable model. With global electric vehicle sales expected to surpass those of conventional cars, EV battery remanufacturing could lead to cost savings and a smaller environmental impact. In 2025, electric vehicle sales are expected to grow by 29.9%, reaching 15.1 million units globally, up from 11.6 million in 2024. This will raise EVs' market share from 13.2% to 16.7%, according to S&P Global Mobility . The IEA forecasts that by 2030, electric cars will account for 65% of global sales. S&P Global Mobility As the EV fleet grows, so will the volume of batteries reaching the end of their life. McKinsey predicts that by 2030, the U.S. will have 40 GWh of used batteries available for resale. That is why remanufacturing is crucial both for the environment and EV battery circularity. predicts When EVs were first introduced, the industry assumed batteries wouldn't fail. As a result, many faulty batteries with healthy parts are being prematurely recycled. Remanufacturing eliminates the need for replacement batteries. Currently, when a battery malfunctions or reaches the end of its useful life, most vehicle owners replace the entire battery pack — often at an exorbitant cost that can exceed $10,000. Entrepreneurs have already spotted a promising opportunity. For example, Tesla cofounder JB Straubel founded Redwood Materials to create a "remanufacturing economy" by extracting and reusing raw materials from used batteries. The startup is on track to hit $200 million in revenue. The EV Battery Remanufacturing Process The EV Battery Remanufacturing Process Unlike traditional manufacturing, remanufacturing is focused on giving old components a second life instead of starting from scratch. With battery packs, this means carefully restoring them to like-new condition. First, each pack undergoes a safety check to ensure it's safe to open, given the high voltage (400 to 800 volts). Next, a thorough inspection looks for any obvious issues, followed by advanced diagnostics with specialized tools. After diagnosing the issue, the battery is carefully disassembled, cleaned, and its components thoroughly inspected. In many cases, only certain parts, such as cells or modules, need replacing, while the rest—when feasible—can be reused. This approach significantly reduces the cost compared to purchasing a brand-new battery pack. Once the damaged parts are replaced with new or refurbished ones, the battery pack is reassembled and undergoes extensive testing to ensure it's safe and fully prepared for use in a new vehicle. This process results in remanufactured battery packs that perform like new. It’s also eco-friendly, reducing waste by reusing components instead of discarding entire packs. The many Benefits of Restoring a Single Battery The many Benefits of Restoring a Single Battery By repairing and reusing battery packs, we lessen the environmental impact of battery disposal. This process helps to minimize the need for new raw materials, including rare minerals like lithium, cobalt, and nickel, which are often extracted in environmentally damaging ways. Remanufacturing also allows us to extend the life of existing batteries. Instead of manufacturing new batteries for every vehicle, we can keep using older packs by replacing only the damaged components, significantly increasing their operational lifespan. This is a win-win for both the environment and consumers, who benefit from lower replacement costs. When a battery is beyond repair, we can recycle it in a way that minimizes environmental damage. Most U.S. facilities use advanced machinery to safely and efficiently extract valuable materials. The materials we extract during the recycling process — such as lithium, cobalt, and nickel — can be sent back to battery manufacturers like LG and Panasonic. By investing in technologies that boost the lifespan of EV batteries, we can pave the way for a future where remanufacturing and recycling are key to a more sustainable transportation system. The future of transport is electric, and with EV remanufacturing, we can make it greener, more efficient, and more affordable for everyone. About the author About the author About the author Aminur Rahman is a leading expert in manufacturing across several industries, including automotive, electric vehicles (EVs), and biotech. He is known for promoting the circular economy and creating sustainable solutions that enhance business performance while benefiting the planet. Through his work, he is helping shape the future of technology toward a more eco-friendly and economically viable model. Aminur Rahman is a leading expert in manufacturing across several industries, including automotive, electric vehicles (EVs), and biotech. He is known for promoting the circular economy and creating sustainable solutions that enhance business performance while benefiting the planet. Through his work, he is helping shape the future of technology toward a more eco-friendly and economically viable model.

Creating a “Remanufacturing Economy” With EV Battery Recycling

The Trials and Triumphs of DPFL Research

Why Smart Power Systems Still Have Big Problems to Solve

Can DPFL Methods Improve Power Grid Predictions? Here’s What We Found

Lost in Linearization: Why Some DPFL Methods Just Can’t Keep Up

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 5. Numerical Evaluations For the numerical analysis, we chose a diverse range of transmission and distribution system models, spanning from 9-bus systems to 1354-bus systems, with 16 distinct configurations. In the following, we first detail the settings for each test case and the methods evaluated. Subsequently, we present the evaluation results with discussions from various perspectives. The open-source code for all the simulations in this paper can be found here [53]. 5.1. Experiment Settings The detailed settings of all test cases are summarized in Table 4. To create the time-series variability in nodal loads and power generations in each test case, a typical daily load curve (in p.u.), shown in Fig. 1, is used as a time-varying scaling factor to change both nodal demand and supply. We also introduce some randomness by adding different fluctuation levels to the scaling factor[2]. Detailed descriptions are provided in the caption of Fig. 1. The resulting supply and demand data are then inputted into AC power flow solvers to generate measurements like nodal voltage, angle, active and reactive power injections, and branch flows. All these measurements are further divided into training and testing datasets. The training dataset is used for DPFL methods to estimate the linear model, while the testing dataset is used to evaluate the linearization errors of DPFL and PPFL methods. The reader is referred to Table 5 for a thorough illustration of the settings for each method, including their predictors, responses, hyperparameters (or the cross-validation ranges[3]), datasets, solvers (when applicable), and additional relevant information. This paper is available on arxiv under CC BY-NC-ND 4.0 Deed (Attribution-Noncommercial-Noderivs 4.0 International) license. [2] Note that for a specific system we use the same scaling curve for all of the methods but across the different systems the randomness is different. [3] In this study, cross-validation is employed for hyperparameter tuning only in cases where the literature does not specify the hyperparameter values. 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 5. Numerical Evaluations For the numerical analysis, we chose a diverse range of transmission and distribution system models, spanning from 9-bus systems to 1354-bus systems, with 16 distinct configurations. In the following, we first detail the settings for each test case and the methods evaluated. Subsequently, we present the evaluation results with discussions from various perspectives. The open-source code for all the simulations in this paper can be found here [53]. 5.1. Experiment Settings The detailed settings of all test cases are summarized in Table 4. To create the time-series variability in nodal loads and power generations in each test case, a typical daily load curve (in p.u.), shown in Fig. 1, is used as a time-varying scaling factor to change both nodal demand and supply. We also introduce some randomness by adding different fluctuation levels to the scaling factor[2]. Detailed descriptions are provided in the caption of Fig. 1. The resulting supply and demand data are then inputted into AC power flow solvers to generate measurements like nodal voltage, angle, active and reactive power injections, and branch flows. All these measurements are further divided into training and testing datasets. The training dataset is used for DPFL methods to estimate the linear model, while the testing dataset is used to evaluate the linearization errors of DPFL and PPFL methods. The reader is referred to Table 5 for a thorough illustration of the settings for each method, including their predictors, responses, hyperparameters (or the cross-validation ranges[3]), datasets, solvers (when applicable), and additional relevant information. 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 [2] Note that for a specific system we use the same scaling curve for all of the methods but across the different systems the randomness is different. [3] In this study, cross-validation is employed for hyperparameter tuning only in cases where the literature does not specify the hyperparameter values.

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