First-order Sensitivity Research Articles

An Experimental and Numerical Study on the Lateral Stiffness Limits of Straddle-Type Monorail Tour-Transit Systems

The development of the straddle-type monorail tour-transit system (MTTS) has received keen attention. Regarding the unspecified regulations on the lateral stiffness limit of steel substructures, which is essential for the design of MTTSs, this work presents a comprehensive assessment of the issue. Firstly, a wind–vehicle–bridge coupling model was established, integrating multibody dynamics and finite element methods. This model was then validated against field test results, considering measured track irregularities and simulated wind loadings as the excitations. Afterwards, a trend analysis and a variance-based sensitivity analysis was performed to investigate the effect of various factors on the dynamic response of the MTTS. Results indicate that the pier height significantly impacts the lateral displacement of the pier top, accounting for 87% of the first-order sensitivity index and 75% of the total sensitivity index. In comparison, the lateral stiffness of track beams contributes over 70% to the maximum responses at the mid-span. Based on this, two responses, the lateral displacement of the pier top and the lateral deflection–span ratio of the track beam, were utilized as evaluation indicators. With the analysis of indicators in terms of lateral acceleration, Sperling index, and lateral wheel force, the limited values for the two indicators were determined as 8.04 mm and L/4200, respectively. These findings can serve as valuable references for future research and designs in this field.

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations: Mathematical Framework and Illustrative Application to the Nordheim–Fuchs Reactor Safety Model

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations: Mathematical Framework and Illustrative Application to the Nordheim–Fuchs Reactor Safety Model

Metacognition bridges experiences and beliefs in sense of agency

Cognitive scientists differentiate the “minimal self” – subjective experiences of agency and ownership in our sensorimotor interactions with the world – from declarative beliefs about the self that are sustained over time. However, it remains an open question how individual sensory experiences of agency are integrated into the belief ofbeing an agent.We administered a sensorimotor task to measure subjects’ (n = 195) propensity to classify stimuli as self-caused and metacognitive monitoring of such judgements, and we compared these behavioral metrics to declarative beliefs about their agency. Subjects who were less sensitive to control cues also reported more negative agency beliefs, though positive beliefs were not clearly correlated with any sensorimotor measure. Importantly, this relationship between first-order sensitivity and declarative beliefs essentially disappears when controlling for metacognitive sensitivity. Results suggest agency beliefs are not related directly to the propensity to make positive agency judgements but are connected through introspective access.

Review of Fourth-Order Maximum Entropy Based Predictive Modeling and Illustrative Application to a Nuclear Reactor Benchmark: II. Best-Estimate Predicted Values and Uncertainties for Model Responses and Parameters

This work continues the review and illustrative application to energy systems of the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” (4th-BERRU-PM) methodology. The 4th-BERRU-PM methodology uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses with respect to model parameters. The 4th-BERRU-PM methodology yields the fourth-order MaxEnt posterior distribution of experimentally measured and computed model responses and parameters in the combined phase space of model responses and parameters. The 4th-BERRU-PM methodology encompasses fourth-order sensitivity analysis (SA) and uncertainty quantification (UQ), which were reviewed in the accompanying work (Part 1), as well as fourth-order data assimilation (DA) and model calibration (MC) capabilities, which will be reviewed and illustrated in this work (Part 2). The applicability of the 4th-BERRU-PM methodology to energy systems is illustrated by using the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark, which is modeled using the linear neutron transport Boltzmann equation, involving 21,976 imprecisely known parameters. This benchmark is representative of “large-scale computations” such as those involved in the modeling of energy systems. The result (“response”) of interest for the PERP benchmark is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. The impact of the high-order sensitivities of the response with respect to the PERP model parameters is quantified for “high-precision” parameters (2% standard deviations) and “typical-precision” parameters (5% standard deviations). Analyzing the best-estimate results with reduced uncertainties for the 1st—through 4th-order moments (mean values, covariance, skewness, and kurtosis) produced by the 4th-BERRU-PM methodology for the PERP benchmark indicates that, even for systems modeled by linear equations (e.g., the PERP benchmark), retaining only first-order sensitivities is insufficient for reliable predictive modeling (including SA, UQ, DA, and MC). At least second-order sensitivities should be retained in order to obtain reliable predictions.

Sensitivity analysis and calibration for a two-dimensional state-space model of metal hydride storage tanks based on experimental data

In this paper, a state-space model of a metal hydride tank is formulated and analyzed in detail. Firstly, a three-dimensional state-space model of the metal hydride tank is simplified by assuming that the tank temperature can be measured. Secondly, the model is completed with a pipe model, which allows to compute the input flow from the pipe pressure that can be easily measured. The proposed model solves the problem that the conventional metal hydride tank models usually take the mass flow rate as input and ignore the pressure as system input. Thirdly, the first-order trajectory sensitivity analysis method is adopted to determine the sensitivity of selected unknown parameters. Latter, the particle swarm optimization algorithm is used to estimate the unknown model parameters from experimental data. Finally, a comparison between experimental data and simulation results demonstrates that the proposed model can reflect the dynamic characteristics of the metal hydride tank.

Adaptive design of experiments to fit surrogate Gaussian process regression models allows fast sensitivity analysis of the input waveform for patient-specific 3D CFD models of liver radioembolization

Background and objectivePatient-specific 3D computational fluid dynamics (CFD) models are increasingly being used to understand and predict transarterial radioembolization procedures used for hepatocellular carcinoma treatment. While sensitivity analyses of these CFD models can help to determine the most impactful input parameters, such analyses are computationally costly. Therefore, we aim to use surrogate modelling to allow relatively cheap sensitivity analysis. As an example, we compute Sobol's sensitivity indices for three input waveform shape parameters. MethodsWe extracted three characteristic shape parameters from our input mass flow rate waveform (peak systolic mass flow rate, heart rate, systolic duration) and defined our 3D input parameter space by varying these parameters within 75 %-125 % of their nominal values. To fit our surrogate model with a minimal number of costly CFD simulations, we developed an adaptive design of experiments (ADOE) algorithm. The ADOE uses 100 Latin hypercube sampled points in 3D input space to define the initial design of experiments (DOE). Subsequently, we re-sample input space with 10,000 Latin Hypercube sampled points and cheaply estimate the outputs using the surrogate model. In each of 27 equivolume bins which divide our input space, we determine the most uncertain prediction of the 10,000 points, compute the true outputs using CFD, and add these points to the DOE. For each ADOE iteration, we calculate Sobol's sensitivity indices, and we continue to add batches of 27 samples to the DOE until the Sobol indices have stabilized. ResultsWe tested our ADOE algorithm on the Ishigami function and showed that we can reliably obtain Sobol's indices with an absolute error <0.1. Applying ADOE to our waveform sensitivity problem, we found that the first-order sensitivity indices were 0.0550, 0.0191 and 0.407 for the peak systolic mass flow rate, heart rate, and the systolic duration, respectively. ConclusionsAlthough the current study was an illustrative case, the ADOE allows reliable sensitivity analysis with a limited number of complex model evaluations, and performs well even when the optimal DOE size is a priori unknown. This enables us to identify the highest-impact input parameters of our model, and other novel, costly models in the future.

OPTIMAL LOCATION OF AN IDEAL TRANSFORMER UPFC MODEL USING OPF FIRST-ORDER SENSITIVITIES

This paper presents a screening technique for greatly reducing the computation involved in determining the optimal location of a unified power flow controller (UPFC) in a power system. The first-order sensitivities of the generation cost with respect to UPFC control parameters are derived. This technique requires running only one optimal power flow (OPF) to obtain UPFC sensitivities for all possible transmission lines. To implement a sensitivity-based screening technique for guidance in optimally locating a single UPFC in a power system, we propose a new UPFC model, wrach consists of an ideal transformer with a complex turns ratio and a variable shunt admittance. In this model, the UPFC control variables do not depend explicitly on UPFC input and output currents and voltages. Accordingly, this model does not require adding extra buses for UPFC input and output terminals. IEEE five-, 14- and 30-bus systems were used to frustrate the technique. Index Terms-Flexible ac transmission systems (FACTS), FACTS location, first-order sensitivity, optimal power flow (OPF), screening technique, unified power flow controller (UPFC), UPFC ideal transformer model, UPFC placement, UPFC uncoupled model. This study focuses on the development of an Ideal Transformer Unified Power Flow Controller (UPFC) model and its integration with Optimal Power Flow (OPF) first-order sensitivities analysis for determining optimal UPFC locations in power systems. The abstract outlines the significance of UPFCs in enhancing power system stability and efficiency by controlling voltage and power flow. It highlights the objectives of the research, including the development of the UPFC model, analysis of OPF first-order sensitivities, and their combined application for screening optimal UPFC locations. The study aims to contribute to the improvement of power system operation and control through strategic UPFC placement. The abstract provides a concise overview of the study, summarizing the key aspects of the research. It outlines the development of an ideal transformer Unified Power Flow Controller (UPFC) model, the analysis of Optimal Power Flow (OPF) first-order sensitivities, and their application in determining optimal locations for UPFC installation in power systems

Design, simulation, and additive manufacturing performance of three lightweight FBG acceleration sensors

Low-weight, high-performance accelerometers are sought-after objectives in aerospace structural design. This paper presents three design schemes for lightweight FBG accelerometers: lattice-based (FBGL), reinforced (FBGR), and lattice-reinforced (FBGL-R) accelerometers. The influence of the design’s filling factor ( F d ) on the first-order frequency and sensitivity in three directions of the accelerometers is investigated. Additionally, the Additive Manufacturing (AM) performance of the designed accelerometers is explored and compared with the original model. The results demonstrate that the first-order natural frequencies of the three design models decrease linearly with increasing F d , with FBGL-R exhibiting a larger adjustable range of low-frequency response under the same mass. FBGL shows the greatest sensitivity variation with F d in all three directions, while FBGR exhibits the least sensitivity variation. Under the same mass fraction, FBGR demonstrates higher sensitivities in all three directions, with FBGL-R's performance falling between the other two designs. AM simulations indicate that the porous structures can reduce the deformation amplitude during the AM of FBG accelerometers. FBGL, FBGR, and FBGL-R exhibit approximately 7% lower deformations compared to the original sensor. The deformation distribution is similar among the four models, with larger residual deformations occurring at the outer edges of the components, while differences in deformation distribution exist within the porous regions.

Isogeometric topology optimization of structures using the overweight approach

In this paper, a 2D isogeometric formulation of the material distribution for structural topology optimization considering minimum weight and local stress constraints using the overweight approach is proposed. The aim of this isogeometric formulation is to provide solutions with high spatial definition using a lower number of design variables in comparison with the formulations previously developed to define the material layout. Despite of this, an important number of local stress constraints has to be considered in the solution of the problem. For this purpose, an Overweight Constraint is used to consider all of them. The structural analysis is performed by means of the Isogeometric Analysis (IGA) and the distribution of material is modeled by means of quadratic B-splines. Moreover, the optimization is addressed by means of the Sequential Linear Programming algorithm (SLP) that is driven by the information provided by a full first-order sensitivity analysis extension of the IGA formulation. Finally, the proposed formulation is tested by means of some benchmark problems, and the results show that the isogeometric formulation provides solutions with high spatial definition. A comparison with a Finite Element Method (FEM) topology optimization formulation is included.

Correction Control Model of L-Index Based on VSC-OPF and BLS Method

With the advancement of artificial intelligence (AI) technology, the real-time measurement and control technology of power systems has also progressed. This paper proposes a correction control model for L-indexes based on voltage stability constrained optimal power flow (VSC-OPF) and a broad learning system (BLS) (BLS-VSC-OPF). This model aims to quickly assess the system’s voltage stability and accurately correct the operation mode when the voltage stability indexes are out of the security range. Firstly, the BLS is used to predict the L-index and to analyze the voltage stability of the power system. Secondly, the approximate first-order sensitivity of the L-index is calculated by the combination of the BLS and the perturbation method. This method solves the problem of the complex sensitivity derivation process in the modeling process of the VSC-OPF model. Meanwhile, when the L-index exceeds the threshold, the BLS and VSC-OPF models are combined to correct this operation mode. The feasibility of the proposed method is verified by the simulation of IEEE-30, IEEE-118, and 1047 bus systems. Finally, the BLS-VSC-OPF model is compared with the linear programming correction model based on BLS (BLS-LPC). The results show that the BLS-VSC-OPF model provides a better correction and control performance.

Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

First-order sensitivities and adjoint analysis are used widely to control the linear stability of unstable flows. Second-order sensitivities have recently helped to increase accuracy. In this paper, a method is presented to calculate arbitrary high-order sensitivities based on Taylor expansions of the incompressible base flow and its eigenproblem around a scalar parameter. For the incompressible Navier–Stokes equations, general expressions for the sensitivities are derived, into which parameter-specific information can be inserted. The computational costs are low since, for all orders, a linear equation system has to be solved, of which the left-hand-side matrix stays constant and thus its preconditioning can be exploited. Two flow scenarios are examined. First, the cylinder flow equations are expanded around the inverse of the Reynolds number, enabling the prediction of the two-dimensional cylinder base flow and its leading eigenvalue as a function of the Reynolds number. This approach computes accurately the base flow and eigenvalue even in the unstable regime, providing, when executed subsequently, a mean to calculate unstable base flows. This case gives a clear introduction into the method and allows us to discuss its constraints regarding convergence behaviour. Second, a small control cylinder is introduced into the domain of the cylinder flow for stabilization. Higher-order sensitivity maps are calculated by modelling the small cylinder with a steady forcing. These maps help to identify stabilizing areas of the flow field for Reynolds numbers within the laminar vortex shedding regime, with the required number of orders increasing as the Reynolds number rises. The results obtained through the proposed method align well with numerically calculated eigenvalues that incorporate the cylinder directly into the grid.

A Systematic Approach of Global Sensitivity Analysis and Its Application to a Model for the Quantification of Resilience of Interconnected Critical Infrastructures

We consider a model for the resilience analysis of interconnected critical infrastructures (ICIs) that describes the dependencies among the subsystems within the ICIs and their time-varying behavior. The model response is a function of uncertain inputs comprising ICIs design parameters and failure magnitudes of vulnerable elements in the system, etc. In this methodological paper, we present a systematic approach based on an innovative blend of methods to perform a sensitivity analysis for identifying the most relevant variables affecting the system resilience at different stages, during a disruptive event. The methods considered include the following: the use of the graphical representation of Cusunoro curves for a visualization of the impact of an input on the resilience metric and an understanding of whether the associated dependence is monotonic, increasing, or decreasing; the introduction of an ensemble of indicators related to different properties of the resilience metric to allow the prioritization of variable importance and avoid false negatives, meaning to regard a variable as non-influential when, instead, it plays a relevant role in the determination of the model response; the calculation of first-order variance-based sensitivity indices to have an appreciation on the relevance of interactions when inputs are independent; and a data approach to visually identify relevant second-order interactions. All the sensitivity methods considered are performed on a provided sample, and do not require additional model evaluations. They allow the analyst to post-process the data to extract, simultaneously, several desirable insights. The systematic approach proposed to apply these methods allows us to identify the model input variables and parameters that are not very relevant, while it enables the identification of the relevant ones which allows prioritizing interventions on the vulnerable elements of the system for its resilience at different stages during a disruptive event. Given the methodological nature of the work, a simplified infrastructure model describing an interconnected gas network and electric power grid is taken as case study: this allows us to show that the approach is straightforward to understand and implement, and the results obtained show the usefulness of the approach in providing meaningful insights that can be used by stakeholders and decision makers to inform strategies for the improvement of system resilience. By the application of the simplified ICIs model to the case study, it is shown that the approach can be straightforwardly implemented to identify the most relevant variables on system resilience and obtain the most important subsystems. The key factors which affect system resilience in multiple initial failures scenarios are found; this allows us to identify the key resilience improvement measurements, and their priorities.

A comprehensive heat transfer prediction model for tubular moving bed heat exchangers using CFD-DEM: Validation and sensitivity analysis

This study established a complete heat transfer prediction model that integrates four heat transfer sub-models, including contact heat conduction, gas film heat conduction, heat radiation, and heat convection, along with an artificial softening correction based on combined approach of computational fluid dynamics and discrete element method (CFD-DEM). The model’s precision was confirmed by experimental results with relative errors below 5 %. The discussion on the heat flux contributions of the above four heat transfer mechanisms shows that with an initial particle temperature of 200℃, tube wall temperature of 30℃, and particle descent velocity of 1.3 mm/s, gas film conduction heat flux is the primary contributor to the total tube wall heat flux at 60.5 %, followed by radiation heat flux (19.5 %), contact conduction heat flux (12.1 %), and convection heat flux (7.9 %). Besides, the presentation of physical fields related to particles and the fluid showcases the model’s capability in elucidating the heat and mass transfer characteristics of the particle–fluid-wall system within tubular MBHEs. Furthermore, a local nominal range sensitivity analysis, along with a study of factors’ influence rules, was performed to quantify the effects of the 8 key model parameters on heat transfer results. The analysis indicates that fluid thermal conductivity is the predominant factor with its nondimensionalized first-order sensitivity coefficient of 54.9 %, succeeded by particle-to-wall angle factor (15.4 %), wall emissivity (14.4 %), gas film thickness (9.6 %), wall thermal conductivity (2.3 %), wall roughness (-1.6 %), particle Young’s modulus (-1.4 %), and particle specific heat capacity (0.4 %). Finally, their impacts on the heat prediction results were presented in detail.

Global Sensitivity Analysis of Factors Influencing the Surface Temperature of Mold during Autoclave Processing.

During the process of forming carbon fiber reinforced plastics (CFRP) in an autoclave, deeply understanding the global sensitivity of factors influencing mold surface temperature is of paramount importance for optimizing large frame-type mold thermally and enhancing curing quality. In this study, the convective heat transfer coefficient (CHTC), the thickness of composite laminates (TCL), the thickness of mold facesheet (TMF), the mold material type (MMT), and the thickness of the auxiliary materials layer (TAL) have been quantitatively assessed for the effects on the mold surface temperature. This assessment was conducted by building the thermal-chemical curing model of composite laminates and utilizing the Sobol global sensitivity analysis (GSA) method. Additionally, the interactions among these factors were investigated to gain a comprehensive understanding of their combined effects. The results show that the sensitivity order of these factors is as follows: CHTC > MMT > TMF > TCL > TAL. Moreover, CHTC, MMT, and TMF are the main factors influencing mold surface temperature, as the sum of their first-order sensitivity indices accounts for over 97.3%. The influence of a single factor is more significant than that of the interaction between factors since the sum of the first-order sensitivity indices of the factors is more than 78.1%. This study will support the development of science-based guidelines for the thermal design of molds and associated heating equipment design.

Analysis of Factors Affecting Vacuum Formation and Drainage in the Siphon-Vacuum Drainage Method for Marine Reclamation

Traditional drainage methods for marine reclamation typically consume large amounts of energy and have a negative environmental impact. The siphon-vacuum drainage method (SVD) automatically forms a vacuum and drains using less energy. It has significant potential for research and application. In this study, a theoretical model is used to calculate the vacuum formation process and drainage rate. Qualitative analysis and global sensitivity analysis were conducted to investigate the effect of various factors in the SVD on vacuum formation and drainage. The qualitative analysis suggests that modifying the length and diameter of the siphon pipe and the thickness of the sealing soil column to increase the siphon rate can improve the vacuum degree and drainage efficiency. Sobol global sensitivity analysis reveals that the sealing soil column thickness is the main factor affecting the vacuum, with a first-order sensitivity index accounting for up to 79.48%. The impact of cylinder diameter and the local resistance coefficient (0.43%) can be almost neglected. A fitting equation for estimating the maximum achievable vacuum is provided. Calculations show that the vacuum formed by the SVD can reach over 80 kPa. This work can help optimize SVD design and advance environmentally friendly marine reclamation projects.

Multi-objective optimization of ice distribution uniformity and startup failure time for cold start of PEMFC catalyst layer based on CTnet and PSO-GA

The distribution of ice in the cathode catalyst layer (CCL) of a proton exchange membrane fuel cell (PEMFC) has a significant impact on cold start. A three-dimensional multiphase numerical model for fuel cell cold start was developed to investigate the first-order finite difference sensitivities of seven selected parameters, namely gas diffusion layer (GDL) porosity, CCL porosity, ionomer volume fraction, inlet temperature, initial membrane water content, catalyst layer (CL) surface area, and initial current, with respect to the uniformity of ice distribution and cold start failure time. Six of these parameters were chosen as inputs for a CNN-Transformers based neural network (CTnet), which was developed using Convolutional Neural Networks (CNN) and the multi-head attention mechanism of Transformers as a surrogate model for cold start. The optimization algorithm PSO-GA was used to find the optimal operating and geometric parameters, aiming to achieve the most uniform ice distribution in the CCL and extend the final cold start failure time. The optimized ice distribution uniformity was found to be 0.204446, showing a 1.1163 % improvement compared to the baseline model. The final cold start failure time was extended by 15.625 % to 185 s compared to the baseline model. The optimized model exhibited a more uniform distribution of ice during cold start, resulting in an extended cold start failure time. This indicates a reduced performance impact on the CCL, thereby improving the durability of the fuel cell. The optimization provides additional time for future auxiliary cold starts, while also enhancing the starting speed of auxiliary cold start through improved ice distribution uniformity.

A Two-Dimensional State-Space Model for Metal Hydride Storage Tanks and Parameter Sensitivity Analysis

Modeling of metal hydride storage tanks remains a challenging issue. In most of the literature, the charge and discharge of metal hydride storage tanks are controlled by the hydrogen flow rate. However, the case of controlling the charge and discharge of metal hydride tanks by pipe pressure has rarely been reported in the literature. Aiming at this problem, this work presents a two-dimensional state-space model that takes the pipe pressure as the system input and assumes that the tank temperature can be measured. Then, the First-order Trajectory Sensitivity Analysis method is used to analyze the parameter sensitivity of several selected unknown parameters. The results of the sensitivity analysis indicate that six parameters: entropy change for desorption ∆Sd, enthalpy change for desorption ∆Hd, activation energy for absorption Ea, activation energy for desorption Ed, plateau flatness coefficient φ and plateau hysteresis coefficient β have higher sensitivity to the model state.

A method of fluorescence molecular tomographic reconstruction via the second-order sensitivity matrix

Fluorescence molecular tomographic (FMT) reconstruction is commonly solved based on the Jacobian matrix, which is a first-order sensitivity matrix. Basically, using the second-order derivatives for iterative reconstruction can help improve the performance of convergence. In this paper, a reconstruction method of FMT based on the reduction of the second-order sensitivity matrix is proposed. In addition, the strategy of detectors rotation is combined into the inverse reconstruction to further improve the reconstruction quality. The reconstructed results demonstrate that the proposed method accelerates the reconstruction with high precision of inverse solutions.

Método sobol para análise de sensibilidade do modelo hydrus na simulação de infiltração de água em depósito aluvionar no semiárido brasileiro

ABSTRACT Simulation of water flow in the soil in semiarid regions is fundamental for managing scarce water resources. Aiming for more accurate results, the sensitivity analysis (SA) of the hydrodynamic parameters involved in the simulations is very relevant. This study aims to evaluate the influence of hydrodynamic parameters on the hydrological response of an alluvial deposit in the dry bed of the Capibaribe River using the Hydrus-1D. A global sensitivity analysis method proposed by Sobol was chosen to evaluate the sensitivity of the soil water infiltration curve. A convergence analysis was carried out to investigate the evolution of the indices with increasing sample size. Values of the sensitivity indices showed gross variations at the beginning of the simulations for the different parameters, but they converged within the set of 10,000 samples. For parameters whose sensitivity indices are closer to 0, it is noted that 1,000 samples (14,000 simulations) were sufficient for the index to converge to their final value. The total and first-order sensitivity index calculated with 10,000 samples present the parameters n and Ks as the most important, being expressly more influential than the others.

Nonlinearity-Adaptive Data-Driven Power Flow Constraint for Distribution Network Optimization

Data-driven power flow (PF) analysis methods have higher practical value to deal with the inaccurate branch parameters in medium and low-voltage distribution networks, which can be easily integrated into the classic optimization framework as a constraint. However, the PF mapping relationship of the measurement data shows stronger nonlinearity, with the high penetration of distributed generation (DG), leading to the low accuracy of the linear PF constraint. To this end, we propose an incomplete dimension lifting (IDL) data-driven PF method for distribution network optimization. The state variables are divided into optimization variables and uncontrollable variables, among them, the optimization variables keep the original linear state space or converted into quadratic space to simplify the optimization solution; and the uncontrollable variables is dimension lifted to fit the nonlinearity of PF. In addition, in order to deal with the problem of gradient direction error caused by nonlinear model overfitting, the first-order sensitivity constraint is involved in parameter regression process. The standard IEEE cases with high penetration of DG demonstrate that the proposed method can achieve higher accurate PF constraint and better optimization results, compared to the existing data-driven PF constrain.