# Adaptive Gradient-Based Optimization Method for Parameter Identification in Power Distribution Network

Authors: Bin Li; Yehai Jiang; Ke Hu; Xiangyi Zhou; Haoran Chen; Shihe Xu; Hao Jiao; Jinming Chen
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Accurate device parameters play a critical role in the calculation and analysis of power distribution networks (PDNs). However, device parameters are always affected by the operating status and influenced by manual entry. Besides, the distribution area of PDN is very wide, which brings more challenges to parameter identification work. Therefore, developing appropriate algorithms for accurately identifying PDN parameters has attracted much more attention from researchers recently. Most of the existing parameter identification algorithms are gradient-free and based on heuristic schemes. Herein, an adaptive gradient-based method is proposed for parameter identification in PDN. The analytical expressions of the gradients of the loss function with respect to the parameters are derived, and an adaptive updating scheme is utilized. By comparing the proposed method and several heuristic algorithms, it is found that the errors in both three criteria via our solution are much lower with a much smoother and more stable convergence of loss function. By further taking a linear transformation of the loss function, the method of this work significantly promotes the parameter identification performance with much lower variance in repeat experiments, indicating that the proposed method in this work achieves a more robust performance to identify PDN parameters. This work gives a practical demonstration by utilizing the gradient-based method for parameter identification of PDN.


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## Body

## 1. Introduction

Obtaining reliable and accurate device parameters is one of the top priorities in the power distribution network (PDN) in many aspects, such as security analysis, system control, state estimation, line loss calculation, power flow calculation, protection setting, and fault analysis [1]. Nevertheless, due to the lack of in-situ measurement techniques, real-time information is hard to obtain directly for PDNs under security and stability situations. Some newly introduced PDN parameters, such as line resistance, line reactance, transformer resistance, transformer reactance, transformer conductance, and transformer electrical susceptance, are in general assumed to be static in a real situation. As a result, it usually leads to poor estimation for parameter identification in PDN [2]. Some new approaches focusing on improving numerical efficiency and error reduction have been developed in many fields of PDN, such as supervisory control and data acquisition, power management unit (PMU), power management unit, and advanced metering infrastructure. These methods can be classified into full-scale approach [3], PSOSR [4], normalized Lagrange multiplier (NLM) test [5], finite-time algorithm (FTA) [6], residual method, sensitivity analysis method, Lagrange multiplier method [7] and Heffron-Phillips method [8]. Furthermore, beneficial to the development of machine learning or deep learning techniques, some smart methods have been proposed recently, such as graph convolution network (GCN) [9], support vector machine (SVM) [10], multihead attention network [11], deep reinforcement learning [12], estimation using synchrophasor data [13], PSCAD simulation [14], and multimodal long short-term memory deep learning [15]. These methods are proved effective with simulation data. However, they are also required to assemble several specialized measuring devices in the grid.The power flow model with identification equations in PDN provides a mathematical approach to solve the problems of lacking required data and measuring devices. It builds relationships between PDN’s parameters and the data which can be obtained easily, such as active power, reaction power, and voltage. Generally, some important parameters are lacking in practice, such as transformer electrical, transformer resistance, transformer conductance, transformer electrical susceptance, line resistance, and line reactance. The ranges of these parameters are always optimized by algorithms combined with the static parameters like active power, reaction power and voltage on the low-voltage side. The calculating values of voltage on the high-voltage can be computed by power flow model with the parameters mentioned before. The residuals between calculated and true values are always used to build loss functions for optimization algorithms. Therefore, the problems of parameter identification can be easily solved by model-free methods, such as least squares (LS) [16] and Markov Chain Monte Carlo (MCMC) [17]. Another kind of methods called heuristic algorithms can also be utilized to identify PDN’s parameters, such as particle swarm optimization (PSO), ant colony, and genetic algorithms (GA) combined with transient measurements [18]. Some algorithms based on global optimization have been published lately for parameter identification in the field of PDN calculation [19], including random search approach (RS) [20], tree-structured Parzen estimator approach (TPE) [21] and simulated annealing (SA) [22], which have shown satisfactory performance and provided novel ideas for PDN analysis. Moreover, some methods based on machine learning and deep learning have been published for parameter identification, such as support vector regression [23], and convolutional neural network (CNN) [24]. These methods are effective for simulation data with the precise values of voltage and voltage phase angle. Nevertheless, only part of the parameters can be obtained.The methods mentioned above for parameter identification can be generally classified as gradient-free methods. In addition, the searching methods in these gradient-free algorithms are largely dependent on the initialization. For example, in the genetic algorithm, to evaluate the designed fitness function, it is first needed to initialize the population via the encoding of the individual. The initialization is usually randomly generated, and different initializations can lead to results with large differences. To tackle these problems in parameter identification, it is crucial to develop more robust and stable numerical methods. In this work, based on the analysis of the physical model in PDN, we derive an analytical theoretical model based on the gradients of the loss function with respect to the parameters. Based on the abovementioned points, this paper mainly has the following contributions:(1)
The analytical expressions of the gradients of the loss function with respect to the parameters to be optimized in PDN are derived in detail, which are rarely studied and neglected by other investigations.(2)
The adaptive scheme for parameter updating based on the loss function gradients is utilized in this work, and it is found that the error during the numerical experiments is remarkably reduced compared with several heuristic algorithms. In addition, the loss function decay during the optimization is much smoother and more stable compared with other algorithms.(3)
The variance in the numerical calculation is much smaller than that of several heuristic algorithms, indicating that the methods proposed in this work are more robust in numerical performance.This paper is organized as follows: Section2 introduces the identification equations of the power flow model in PDN and proposes the gradient-based optimization method. The experimental data and calculation details are given in Section 3. The results and relevant discussions are given in Section 4. Finally, Section 5 gives a brief conclusion.


## 2. Materials and Methods

### 2.1. Power Flow Model Calculation

The basic theory of analysis in PDN can be found in ref [17, 25]. To simplify the computation, the three-phase is assumed to be balanced as the premise for calculating the power flow in this work. The schematic diagram of the power-flow calculation circuit model is shown in Figure 1.Figure 1
Power flow calculation circuit model.In Figure1, Pd,Qd and Ud represent the active power, reaction power, and voltage on the high-voltage side of the transformers at bus D, respectively. These three parameters can be obtained directly by real-time measurements. Other parameters, such as transformer electrical Rd, transformer resistance Xd, transformer conductance Gd, transformer electrical susceptance Bd, line resistance Rcd and line reactance Xcd, are in general hard to detect in PDN calculation, and satisfy equations (1)–(3):(1)Pd=PLd+PLd2+QLd2ULd2RdT+ULd2Gd2,(2)Qd=QLd+PLd2+QLd2ULd2XdT+ULd2Bd2,(3)Ud=ULd+ΔUdT2+δUdT2,where ΔUdT and δUdT in equation (3) are the longitudinal and transverse components of the transformer impedance voltage drop at bus D in V. PLd,QLd and ULd represent the active power, reaction power, and voltage on the low-voltage side of the transformers at bus D, respectively. ΔUdT and δUdT can be expressed by equations (4) and (5), respectively.(4)ΔUdT=PLdRdT+QLdXdTULd,(5)δUdT=PLdXdT−QLdRdTULd.The equation of busC can be expressed as equations (6)∼(8):(6)Uc=Ud+ΔUcdT2+δUcdT2,(7)ΔUcdT=PdRcdT+QdXcdTULd,(8)δUcdT=PdXcdT−QdRcdTUd,where ΔUcdT and δUcdT in equation (6) are the longitudinal and transverse components of the transformer impedance voltage drop at bus C in V. Once we have the above quantities, the final result is calculated by the following equation.(9)fc=Ud+ΔUcdT2+δUcdT2.The parameters in the line and transformer can be calculated based on equations (1)∼(9) with the measured power and voltage data.


### 2.2. Gradient-Based Algorithm for Parameter Identification in PDN

To utilize the gradient-based algorithms for PDN calculations, first the surrogate function (or, namely, the objective function) should be defined. As mentioned above, the inputs of PDN can be defined as:X=PLd,i,QLd,i,ULd,i representing the set of the active power, reaction power, and voltage on the high-voltage side of the transformers, and the subscript i represents the sample points for i=1,2,…,N. While other parameters, namely, Rd,Xd,Gd,Bd,Rcd and Xcd are seen as the initialized parameters, which are needed to be optimized. Herein, we consider a linear regression of the target value Y with the combination of the inputs as follows:(10)Y=θ1X+θ0+η,where θ0 represents the bias term and η represents the noise term. In this work, we assume that the noise term is subject to the distribution where the mean value and variance are 0 and 1, respectively, namely, η∼N0,1. Therefore, the mean value and variance of the target value Y are(11)EY=Eθ1X+θ0+η=θ1X+θ0,(12)VarY=Varθ1X+θ0+η=1.Therefore, the target values are subject to the distribution where its mean value isθ1X+θ0 and the variance is 1. Then the probability of the sample point xi,yi is(13)pyi|xi=exp−yi−θ1xi+θ022.The likelihood probability of the whole sample sets is(14)LX,Y=∏i=1Npyi|xi=∏i=1Nexp−yi−θ1xi+θ022.Take the logarithm of the likelihood function above and then, we obtain(15)lX,Y=logLX,Y=−12∑i=1Nyi−θ1xi+θ02.Extend the above logarithm of the likelihood function, and the loss function in this work is designed as(16)loss=12N∑i=1Nyi−y⌢i2=12N∑i=1Nθ1,ifc,i+θ0,i−U⌢c,i2,where yi and y⌢i represents the calculation value and true value, respectively. Herein, we also introduce a linear transform with parameter θ1,i and θ0,i,U⌢c,i represents the true value of voltage at the on the high-voltage side.


#### 2.2.1. Adaptive Gradient-Based Optimization Methods

Up to now, we have known that the optimization mission in this work is to minimize the loss function mentioned above with respect to the following six parameters, namely,Rd,Xd,Gd,Bd,Rcd and Xcd:(17)minimize lossθ∈ℝd=lossRd,Xd,Gd,Bd,Rcd,Xcd.To utilize the gradient-based optimization methods, it is generally needed to calculate the gradient of the loss function with respect to the parameters, namely,gt=∇θlossθ. In addition, the historical information during the optimization in each step can be utilized to accelerate the optimization, obtaining the first-order and second-order moments as follows:(18)mt=ϕg1,g2,…,gt,(19)vt=ψg1,g2,…,gt.Then, the parameters are updated with the following rule:(20)θt+1=θt−1vt+εmt,where ε represents a smooth term to prevent the denominator equaling to zero. In equation (20), if we ignore the second-order moment vt and set it as vt=I2,mt=ηgt (where η is often known as the learning rate), then the optimization scheme is recovered as the stochastic gradient descent approach. The convergence of the stochastic gradient descent approach is usually slow and it is easy to oscillate at the saddle point. In addition, different parameters should adopt different updating schemes, which means some parameters may not update so frequently during the optimization, thus, it is expected to utilize a relatively large step to speed up the convergence. Oppositely, for those parameters updating frequently, generally it is needed to take a small step for achieving a more stable optimization process. Based on these analyses, the following update scheme is utilized:(21)vt=diag∑i=1tgi,12,∑i=1tgi,22,…,∑i=1tgi,d2,where vt∈ℝd×d is a diagonal matrix. Equation (21) indicates that for frequently updating parameters before the moment t, its second-order momentum is relatively large and results in a small learning rate. The learning rate in the gradient-based optimization method is also called the hyperparameter, which can be dynamically changed with an appropriate schedule. To test the influence of hyperparameters such as the learning rate on the performance of parameter identification in PDN. To utilize the gradient-based method to optimize six parameters, namely, Rd,Xd,Gd,Bd,Rcd and Xcd, the gradient of the loss function with respect to these parameters needs to be derived in the next section.


#### 2.2.2. Gradients of the Loss Function Based on Chain-Rule

Once we have the loss function, the next step is to calculate the gradients of the loss function with respect to the six parameters to be optimized. With the help of the chain-rule, the gradients can be derived. First,∂loss/∂Rd and ∂loss/∂Xd are given as follows:(22)∂loss∂Rd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Rd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Rd,(23)∂loss∂Xd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Xd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Xd,where the derivatives ∂loss/∂fc⋅∂fc/∂Ud is shown inthe both above two equations, and it is given as follows:(24)∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2.Also, the derivatives∂Ud/∂ΔUdT⋅∂ΔUdT/∂Rd and ∂Ud/∂δUdT⋅∂δUdT/∂Rd are derived as(25)∂Ud∂ΔUdT⋅∂ΔUdT∂Rd=ULd,i+ΔUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅PLd,iULd,i,(26)∂Ud∂δUdT⋅∂δUdT∂Rd=−δUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅QLd,iULd,i.Substituting (19)–(21) into (17) and (18), then we obtain(27)∂loss∂Rd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT−QLd,iδUd,iTULd,i,(28)∂loss∂Xd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT+QLd,iδUd,iTULd,i.Similar to the derivation of (22) and (23), the gradient of the loss function with respect to the other parameters is(29)∂loss∂Gd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Pd⋅∂Pd∂Gd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Pd⋅∂Pd∂Gd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Gd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Rcd,iTULd,i+δUcd,iT⋅Xcd,iTUd,i,(30)∂loss∂Bd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Qd⋅∂Qd∂Bd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Qd⋅∂Qd∂Bd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Bd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Xcd,iTULd,i−δUcd,iT⋅Rcd,iTUd,i,(31)∂loss∂Rcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Rcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Rcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Pd,i/ULd,i−δUcd,iT⋅Qd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2,(32)∂loss∂Xcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Xcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Xcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Qd,i/ULd,i−δUcd,iT⋅Pd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2.Now, from (17)–(27), we have the six gradients of the loss function with respect to the parameters to be optimized.Once we have the above gradients of the loss function with respect to the parameters, then the gradient-based optimization can be implemented. The pseudo-code of the optimization method of this work is shown in Algorithm1:Algorithm 1: Adaptive gradient-based optimization methods.
Input: θ0 initial parametersfθ objective function to be optimizedβ1,β2 decay rates for moment estimatesm0 initialized first-order momentυ0 initialized second-order momentt time stepη learning rateWhileθt is not converged dot←t+1gt←∇θftθt−1 gradient w.r.t parameters at time step tmt←β1⋅mt−1+1−β1⋅gt update first-order momentυt←β2⋅υt−1+1−β2⋅gt2 update second-order momentm^t←mt/1−β1t biased-corrected first-order momentυ^t←υt/1−β2t biased-corrected second-order momentθt←θt−1−ηm^t/υ^t+εEnd whileReturnθt


### 2.3. Evaluation Functions of the Algorithm

The underlying three functions are employed to estimate the performance of the algorithm proposed in this work:(1)
Mean absolute error (MAE):(33)MAE=1N∑i=1Nyi−y⌢i.(2)
Root mean square error (RMSE):(34)RMSE=1N∑i=1Nyi−y⌢i2.(3)
Mean absolute percentage error (MAPE):(35)MAPE=100%N∑i=1Nyi−y⌢iyi,whereyi and y⌢i represent the ground true value and prediction value, respectively.


## 2.1. Power Flow Model Calculation

The basic theory of analysis in PDN can be found in ref [17, 25]. To simplify the computation, the three-phase is assumed to be balanced as the premise for calculating the power flow in this work. The schematic diagram of the power-flow calculation circuit model is shown in Figure 1.Figure 1
Power flow calculation circuit model.In Figure1, Pd,Qd and Ud represent the active power, reaction power, and voltage on the high-voltage side of the transformers at bus D, respectively. These three parameters can be obtained directly by real-time measurements. Other parameters, such as transformer electrical Rd, transformer resistance Xd, transformer conductance Gd, transformer electrical susceptance Bd, line resistance Rcd and line reactance Xcd, are in general hard to detect in PDN calculation, and satisfy equations (1)–(3):(1)Pd=PLd+PLd2+QLd2ULd2RdT+ULd2Gd2,(2)Qd=QLd+PLd2+QLd2ULd2XdT+ULd2Bd2,(3)Ud=ULd+ΔUdT2+δUdT2,where ΔUdT and δUdT in equation (3) are the longitudinal and transverse components of the transformer impedance voltage drop at bus D in V. PLd,QLd and ULd represent the active power, reaction power, and voltage on the low-voltage side of the transformers at bus D, respectively. ΔUdT and δUdT can be expressed by equations (4) and (5), respectively.(4)ΔUdT=PLdRdT+QLdXdTULd,(5)δUdT=PLdXdT−QLdRdTULd.The equation of busC can be expressed as equations (6)∼(8):(6)Uc=Ud+ΔUcdT2+δUcdT2,(7)ΔUcdT=PdRcdT+QdXcdTULd,(8)δUcdT=PdXcdT−QdRcdTUd,where ΔUcdT and δUcdT in equation (6) are the longitudinal and transverse components of the transformer impedance voltage drop at bus C in V. Once we have the above quantities, the final result is calculated by the following equation.(9)fc=Ud+ΔUcdT2+δUcdT2.The parameters in the line and transformer can be calculated based on equations (1)∼(9) with the measured power and voltage data.


## 2.2. Gradient-Based Algorithm for Parameter Identification in PDN

To utilize the gradient-based algorithms for PDN calculations, first the surrogate function (or, namely, the objective function) should be defined. As mentioned above, the inputs of PDN can be defined as:X=PLd,i,QLd,i,ULd,i representing the set of the active power, reaction power, and voltage on the high-voltage side of the transformers, and the subscript i represents the sample points for i=1,2,…,N. While other parameters, namely, Rd,Xd,Gd,Bd,Rcd and Xcd are seen as the initialized parameters, which are needed to be optimized. Herein, we consider a linear regression of the target value Y with the combination of the inputs as follows:(10)Y=θ1X+θ0+η,where θ0 represents the bias term and η represents the noise term. In this work, we assume that the noise term is subject to the distribution where the mean value and variance are 0 and 1, respectively, namely, η∼N0,1. Therefore, the mean value and variance of the target value Y are(11)EY=Eθ1X+θ0+η=θ1X+θ0,(12)VarY=Varθ1X+θ0+η=1.Therefore, the target values are subject to the distribution where its mean value isθ1X+θ0 and the variance is 1. Then the probability of the sample point xi,yi is(13)pyi|xi=exp−yi−θ1xi+θ022.The likelihood probability of the whole sample sets is(14)LX,Y=∏i=1Npyi|xi=∏i=1Nexp−yi−θ1xi+θ022.Take the logarithm of the likelihood function above and then, we obtain(15)lX,Y=logLX,Y=−12∑i=1Nyi−θ1xi+θ02.Extend the above logarithm of the likelihood function, and the loss function in this work is designed as(16)loss=12N∑i=1Nyi−y⌢i2=12N∑i=1Nθ1,ifc,i+θ0,i−U⌢c,i2,where yi and y⌢i represents the calculation value and true value, respectively. Herein, we also introduce a linear transform with parameter θ1,i and θ0,i,U⌢c,i represents the true value of voltage at the on the high-voltage side.


### 2.2.1. Adaptive Gradient-Based Optimization Methods

Up to now, we have known that the optimization mission in this work is to minimize the loss function mentioned above with respect to the following six parameters, namely,Rd,Xd,Gd,Bd,Rcd and Xcd:(17)minimize lossθ∈ℝd=lossRd,Xd,Gd,Bd,Rcd,Xcd.To utilize the gradient-based optimization methods, it is generally needed to calculate the gradient of the loss function with respect to the parameters, namely,gt=∇θlossθ. In addition, the historical information during the optimization in each step can be utilized to accelerate the optimization, obtaining the first-order and second-order moments as follows:(18)mt=ϕg1,g2,…,gt,(19)vt=ψg1,g2,…,gt.Then, the parameters are updated with the following rule:(20)θt+1=θt−1vt+εmt,where ε represents a smooth term to prevent the denominator equaling to zero. In equation (20), if we ignore the second-order moment vt and set it as vt=I2,mt=ηgt (where η is often known as the learning rate), then the optimization scheme is recovered as the stochastic gradient descent approach. The convergence of the stochastic gradient descent approach is usually slow and it is easy to oscillate at the saddle point. In addition, different parameters should adopt different updating schemes, which means some parameters may not update so frequently during the optimization, thus, it is expected to utilize a relatively large step to speed up the convergence. Oppositely, for those parameters updating frequently, generally it is needed to take a small step for achieving a more stable optimization process. Based on these analyses, the following update scheme is utilized:(21)vt=diag∑i=1tgi,12,∑i=1tgi,22,…,∑i=1tgi,d2,where vt∈ℝd×d is a diagonal matrix. Equation (21) indicates that for frequently updating parameters before the moment t, its second-order momentum is relatively large and results in a small learning rate. The learning rate in the gradient-based optimization method is also called the hyperparameter, which can be dynamically changed with an appropriate schedule. To test the influence of hyperparameters such as the learning rate on the performance of parameter identification in PDN. To utilize the gradient-based method to optimize six parameters, namely, Rd,Xd,Gd,Bd,Rcd and Xcd, the gradient of the loss function with respect to these parameters needs to be derived in the next section.


### 2.2.2. Gradients of the Loss Function Based on Chain-Rule

Once we have the loss function, the next step is to calculate the gradients of the loss function with respect to the six parameters to be optimized. With the help of the chain-rule, the gradients can be derived. First,∂loss/∂Rd and ∂loss/∂Xd are given as follows:(22)∂loss∂Rd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Rd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Rd,(23)∂loss∂Xd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Xd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Xd,where the derivatives ∂loss/∂fc⋅∂fc/∂Ud is shown inthe both above two equations, and it is given as follows:(24)∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2.Also, the derivatives∂Ud/∂ΔUdT⋅∂ΔUdT/∂Rd and ∂Ud/∂δUdT⋅∂δUdT/∂Rd are derived as(25)∂Ud∂ΔUdT⋅∂ΔUdT∂Rd=ULd,i+ΔUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅PLd,iULd,i,(26)∂Ud∂δUdT⋅∂δUdT∂Rd=−δUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅QLd,iULd,i.Substituting (19)–(21) into (17) and (18), then we obtain(27)∂loss∂Rd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT−QLd,iδUd,iTULd,i,(28)∂loss∂Xd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT+QLd,iδUd,iTULd,i.Similar to the derivation of (22) and (23), the gradient of the loss function with respect to the other parameters is(29)∂loss∂Gd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Pd⋅∂Pd∂Gd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Pd⋅∂Pd∂Gd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Gd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Rcd,iTULd,i+δUcd,iT⋅Xcd,iTUd,i,(30)∂loss∂Bd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Qd⋅∂Qd∂Bd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Qd⋅∂Qd∂Bd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Bd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Xcd,iTULd,i−δUcd,iT⋅Rcd,iTUd,i,(31)∂loss∂Rcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Rcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Rcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Pd,i/ULd,i−δUcd,iT⋅Qd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2,(32)∂loss∂Xcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Xcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Xcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Qd,i/ULd,i−δUcd,iT⋅Pd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2.Now, from (17)–(27), we have the six gradients of the loss function with respect to the parameters to be optimized.Once we have the above gradients of the loss function with respect to the parameters, then the gradient-based optimization can be implemented. The pseudo-code of the optimization method of this work is shown in Algorithm1:Algorithm 1: Adaptive gradient-based optimization methods.
Input: θ0 initial parametersfθ objective function to be optimizedβ1,β2 decay rates for moment estimatesm0 initialized first-order momentυ0 initialized second-order momentt time stepη learning rateWhileθt is not converged dot←t+1gt←∇θftθt−1 gradient w.r.t parameters at time step tmt←β1⋅mt−1+1−β1⋅gt update first-order momentυt←β2⋅υt−1+1−β2⋅gt2 update second-order momentm^t←mt/1−β1t biased-corrected first-order momentυ^t←υt/1−β2t biased-corrected second-order momentθt←θt−1−ηm^t/υ^t+εEnd whileReturnθt


## 2.2.1. Adaptive Gradient-Based Optimization Methods

Up to now, we have known that the optimization mission in this work is to minimize the loss function mentioned above with respect to the following six parameters, namely,Rd,Xd,Gd,Bd,Rcd and Xcd:(17)minimize lossθ∈ℝd=lossRd,Xd,Gd,Bd,Rcd,Xcd.To utilize the gradient-based optimization methods, it is generally needed to calculate the gradient of the loss function with respect to the parameters, namely,gt=∇θlossθ. In addition, the historical information during the optimization in each step can be utilized to accelerate the optimization, obtaining the first-order and second-order moments as follows:(18)mt=ϕg1,g2,…,gt,(19)vt=ψg1,g2,…,gt.Then, the parameters are updated with the following rule:(20)θt+1=θt−1vt+εmt,where ε represents a smooth term to prevent the denominator equaling to zero. In equation (20), if we ignore the second-order moment vt and set it as vt=I2,mt=ηgt (where η is often known as the learning rate), then the optimization scheme is recovered as the stochastic gradient descent approach. The convergence of the stochastic gradient descent approach is usually slow and it is easy to oscillate at the saddle point. In addition, different parameters should adopt different updating schemes, which means some parameters may not update so frequently during the optimization, thus, it is expected to utilize a relatively large step to speed up the convergence. Oppositely, for those parameters updating frequently, generally it is needed to take a small step for achieving a more stable optimization process. Based on these analyses, the following update scheme is utilized:(21)vt=diag∑i=1tgi,12,∑i=1tgi,22,…,∑i=1tgi,d2,where vt∈ℝd×d is a diagonal matrix. Equation (21) indicates that for frequently updating parameters before the moment t, its second-order momentum is relatively large and results in a small learning rate. The learning rate in the gradient-based optimization method is also called the hyperparameter, which can be dynamically changed with an appropriate schedule. To test the influence of hyperparameters such as the learning rate on the performance of parameter identification in PDN. To utilize the gradient-based method to optimize six parameters, namely, Rd,Xd,Gd,Bd,Rcd and Xcd, the gradient of the loss function with respect to these parameters needs to be derived in the next section.


## 2.2.2. Gradients of the Loss Function Based on Chain-Rule

Once we have the loss function, the next step is to calculate the gradients of the loss function with respect to the six parameters to be optimized. With the help of the chain-rule, the gradients can be derived. First,∂loss/∂Rd and ∂loss/∂Xd are given as follows:(22)∂loss∂Rd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Rd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Rd,(23)∂loss∂Xd=∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂ΔUdT⋅∂ΔUdT∂Xd+∂loss∂fc⋅∂fc∂Ud⋅∂Ud∂δUdT⋅∂δUdT∂Xd,where the derivatives ∂loss/∂fc⋅∂fc/∂Ud is shown inthe both above two equations, and it is given as follows:(24)∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2.Also, the derivatives∂Ud/∂ΔUdT⋅∂ΔUdT/∂Rd and ∂Ud/∂δUdT⋅∂δUdT/∂Rd are derived as(25)∂Ud∂ΔUdT⋅∂ΔUdT∂Rd=ULd,i+ΔUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅PLd,iULd,i,(26)∂Ud∂δUdT⋅∂δUdT∂Rd=−δUd,iTULd,i+ΔUd,iT2+δUd,iT2⋅QLd,iULd,i.Substituting (19)–(21) into (17) and (18), then we obtain(27)∂loss∂Rd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT−QLd,iδUd,iTULd,i,(28)∂loss∂Xd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅Ud,i+ΔUcd,iTUd,i+ΔUcd,iT2+δUcd,iT2⋅δUd,iTULd,i+ΔUd,iT2+δUd,iT2PLd,iULd,i+ΔUd,iT+QLd,iδUd,iTULd,i.Similar to the derivation of (22) and (23), the gradient of the loss function with respect to the other parameters is(29)∂loss∂Gd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Pd⋅∂Pd∂Gd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Pd⋅∂Pd∂Gd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Gd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Rcd,iTULd,i+δUcd,iT⋅Xcd,iTUd,i,(30)∂loss∂Bd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Qd⋅∂Qd∂Bd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Qd⋅∂Qd∂Bd=∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,iN⋅2ULd,i2Bd,iUd,i+ΔUcd,iT2+δUcd,iT2⋅Ud,i+ΔUcd,iT⋅Xcd,iTULd,i−δUcd,iT⋅Rcd,iTUd,i,(31)∂loss∂Rcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Rcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Rcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Pd,i/ULd,i−δUcd,iT⋅Qd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2,(32)∂loss∂Xcd=∂loss∂fc⋅∂fc∂ΔUcdT⋅∂ΔUcdT∂Xcd+∂loss∂fc⋅∂fc∂δUcdT⋅∂δUcdT∂Xcd=1N∑i=1Nθ1,iθ1,ifc,i+θ0,i−U⌢c,i⋅Ud,i+ΔUcd,iT⋅Qd,i/ULd,i−δUcd,iT⋅Pd,i/Ud,iUd,i+ΔUcd,iT2+δUcd,iT2.Now, from (17)–(27), we have the six gradients of the loss function with respect to the parameters to be optimized.Once we have the above gradients of the loss function with respect to the parameters, then the gradient-based optimization can be implemented. The pseudo-code of the optimization method of this work is shown in Algorithm1:Algorithm 1: Adaptive gradient-based optimization methods.
Input: θ0 initial parametersfθ objective function to be optimizedβ1,β2 decay rates for moment estimatesm0 initialized first-order momentυ0 initialized second-order momentt time stepη learning rateWhileθt is not converged dot←t+1gt←∇θftθt−1 gradient w.r.t parameters at time step tmt←β1⋅mt−1+1−β1⋅gt update first-order momentυt←β2⋅υt−1+1−β2⋅gt2 update second-order momentm^t←mt/1−β1t biased-corrected first-order momentυ^t←υt/1−β2t biased-corrected second-order momentθt←θt−1−ηm^t/υ^t+εEnd whileReturnθt


## 2.3. Evaluation Functions of the Algorithm

The underlying three functions are employed to estimate the performance of the algorithm proposed in this work:(1)
Mean absolute error (MAE):(33)MAE=1N∑i=1Nyi−y⌢i.(2)
Root mean square error (RMSE):(34)RMSE=1N∑i=1Nyi−y⌢i2.(3)
Mean absolute percentage error (MAPE):(35)MAPE=100%N∑i=1Nyi−y⌢iyi,whereyi and y⌢i represent the ground true value and prediction value, respectively.


## 3. Data and Calculation Details

### 3.1. Raw Dataset Description

In this paper, 1499 samples in the raw dataset were collected by SCADA with a sampling period of 15 minutes [25, 26]. The three-phase first section voltages on the high-voltage side (denoted as Ua,Ub and Uc) are shown in Figure 2, and the low-voltage sides of them (denoted as ua,ub, and uc) are displayed in Figure 3.Figure 2
Ua, Ub, and Uc on the high-voltage side.Figure 3
ua, ub, and uc on the high-voltage side.It can be found in Figures2 and 3 that the high-voltage sides in the dataset are closed to the three-phase balance, and this dataset satisfies the requirements of the equations in Section 2.1. In addition, the active power (denoted as Pa,Pb, and Pc) and reactive power (denoted as Qa,Qb, and Qc) of three-phase on the low voltage side are shown in Figures 4 and 5, respectively.Figure 4
Pa, Pb, and Pc on the low voltage side.Figure 5
Qa, Qb, and Qc on the low voltage side.The trend of changes of active power and reactive power on the low voltage side is consistent in Figures4 and 5, and it indicates that the samples in this dataset are stable and can be used to perform parameter identification.


### 3.2. Evaluation and Calculation

In this paper, 75% of the samples (1124) are split randomly as a training set to identify PDN’s parameters. The best parameters are used to calculate voltage per unit inC bus (denoted as Ucal) by a power flow model. After that, the rest of the 25% samples (375) are used to evaluate the performance of parameter identification as a test set through the three metrics as equations (28)–(30). Instead of directly calculating these metrics, the linear regression should be applied in this paper, the values of Uc and Ucal are regarded as dependent variable and independent variable, respectively. The output values of linear regression are signed as Ucal∗, and the final evaluations of parameter identification are gained between Uc and Ucal∗ as equation (31):(36)Ucal∗=aUcal+b,where a and b are denoted as the slope and bias of linear regression, and they can also be optimized like other parameters by MCMC and GA. In the following discussion, the parameters of linear regression optimized by MCMC and GA are signed as MCMC-LR and GA-LR, respectively. The upper bounds and lower bounds of MCMC and GA should be determined before parameter identification, and they are listed in Table 1:Table 1
The upper and lower bounds of MCMC and GA.
Parameter nameAbbreviationUpper boundLower boundUnitLine resistanceRcd0.50.005Ω/kmLine reactanceXcd0.50.005Ω/kmTransformer resistanceXd205Ω/kmTransformer reactanceRd100.8Ω/kmTransformer conductanceGd8e-64e-6STransformer electrical susceptanceBd8e-52e-5SSlope of linear regressiona−55∗Bias of linear regressionb5001000∗To avoid the impact of randomness on GA and MCMC, each method is repeated 25 times to guarantee the correctness and stability of results.


## 3.1. Raw Dataset Description

In this paper, 1499 samples in the raw dataset were collected by SCADA with a sampling period of 15 minutes [25, 26]. The three-phase first section voltages on the high-voltage side (denoted as Ua,Ub and Uc) are shown in Figure 2, and the low-voltage sides of them (denoted as ua,ub, and uc) are displayed in Figure 3.Figure 2
Ua, Ub, and Uc on the high-voltage side.Figure 3
ua, ub, and uc on the high-voltage side.It can be found in Figures2 and 3 that the high-voltage sides in the dataset are closed to the three-phase balance, and this dataset satisfies the requirements of the equations in Section 2.1. In addition, the active power (denoted as Pa,Pb, and Pc) and reactive power (denoted as Qa,Qb, and Qc) of three-phase on the low voltage side are shown in Figures 4 and 5, respectively.Figure 4
Pa, Pb, and Pc on the low voltage side.Figure 5
Qa, Qb, and Qc on the low voltage side.The trend of changes of active power and reactive power on the low voltage side is consistent in Figures4 and 5, and it indicates that the samples in this dataset are stable and can be used to perform parameter identification.


## 3.2. Evaluation and Calculation

In this paper, 75% of the samples (1124) are split randomly as a training set to identify PDN’s parameters. The best parameters are used to calculate voltage per unit inC bus (denoted as Ucal) by a power flow model. After that, the rest of the 25% samples (375) are used to evaluate the performance of parameter identification as a test set through the three metrics as equations (28)–(30). Instead of directly calculating these metrics, the linear regression should be applied in this paper, the values of Uc and Ucal are regarded as dependent variable and independent variable, respectively. The output values of linear regression are signed as Ucal∗, and the final evaluations of parameter identification are gained between Uc and Ucal∗ as equation (31):(36)Ucal∗=aUcal+b,where a and b are denoted as the slope and bias of linear regression, and they can also be optimized like other parameters by MCMC and GA. In the following discussion, the parameters of linear regression optimized by MCMC and GA are signed as MCMC-LR and GA-LR, respectively. The upper bounds and lower bounds of MCMC and GA should be determined before parameter identification, and they are listed in Table 1:Table 1
The upper and lower bounds of MCMC and GA.
Parameter nameAbbreviationUpper boundLower boundUnitLine resistanceRcd0.50.005Ω/kmLine reactanceXcd0.50.005Ω/kmTransformer resistanceXd205Ω/kmTransformer reactanceRd100.8Ω/kmTransformer conductanceGd8e-64e-6STransformer electrical susceptanceBd8e-52e-5SSlope of linear regressiona−55∗Bias of linear regressionb5001000∗To avoid the impact of randomness on GA and MCMC, each method is repeated 25 times to guarantee the correctness and stability of results.


## 4. Results and Discussion

The parameter identification results of AGBO and three SMBO algorithms based on mean square error betweenUc and Ucal are discussed first in this section. The prior weight and number of started jobs are set as 1 and 20 for the TPE method. The rate of reduction in simulated annealing is 0.1 as a default value [25]. The learning rate and weight decay are 5e-4 and 0 in AGBO. The maximum iteration step is 1000. Their prediction performances are shown in Table 2, Figures 6 and 7, respectively.Table 2
The results of parameter identification with the loss function of mean square error.
AlgorithmMAERMSEMAPERS65.95067.5001.078TPE64.64866.3401.056SA66.00367.5501.078AGBO∗64.38466.0911.052Figure 6
The scatter plot ofUc and Ucal by RS (a), TPE (b), SA (c), and AGBO (d).
(a)(b)(c)(d)Figure 7
The line plot ofUc and Ucal by RS (a), TPE (b), SA (c),and AGBO (d).
(a)(b)(c)(d)It can be found in Table2 that AGBO has the lowest values in MAE, RMSE, and MAPE compared with the results of RS, TPE, and SA. However, their results only have minor differences, and Figures 6 and 7 also show that the prediction results by these algorithms do not have remarkable differences. The reason can be attributed to the fact that calculating mean square error of Uc and Ucal neglecting the statistical relationship between them and has a negative impact on parameter identification. Therefore, based on the previous study [26], the line transformation can be implemented to Ucal before calculating the loss function. The parameters of line transformation are optimized with other PDN parameters simultaneously, and their results are displayed in Table 3, Figures 8 and 9.Table 3
The results of parameter identification with linear transformation.
AlgorithmMAERMSEMAPERS-LR6.3387.8590.104TPE-LR6.0007.6160.098SA-LR7.4509.3280.122AGBO-LR5.3906.7410.088Figure 8
The scatter plot ofUc and Ucal by RS (a), TPE (b), SA (c) , and AGBO (d) with linear transformation.
(a)(b)(c)(d)Figure 9
The line plot ofUc and Ucal by RS (a), TPE (b), SA (c), and AGBO (d) with linear transformation.
(a)(b)(c)(d)All algorithms perform better in Table3 than the results in Table 2, which indicates the statistical property between Uc and Ucal, such as linear relationship, has contributed to identifying PDN’s parameters. The prediction errors in Table 3 are almost one-tenth of those in Table 2. Moreover, it can be found that AGBO-LR has a significantly better prediction performance evaluated by all of the three metrics compared with RS-LR, TPE-LR, and SA-LR in Table 3, and the scatter points are closer to a line of 45 degrees than the others, which mean the prediction error of AGBO-LR is lower.Learning rate and weight decay are two important hyperparameters in the optimization proposed in this work. Therefore, the performance for various values of these parameters is also investigated. The value of weight decay is set as 0 first, and the performances of AGBO with different learning rates are displayed in Table4:Table 4
The performance of AGBO under different learning rates.
Learning rateMAERMSEMAPE1e-550.28251.3000.8225e-515.12716.9530.2471e-46.9878.6930.1145e-45.6527.0970.0921e-35.3976.7710.0885e-35.2736.6450.0861e-210.85912.6700.1775e-214.10716.0920.231It is found in Table4 that the variation of the learning rate has a large influence on the performance of optimization, and the value of the learning rate between 5e-3 and 5e-4 is suggested in this paper. The influence of weight decay on AGBO is investigated subsequently with the optimal value of learning rate; the results are listed in Table 5:Table 5
The performance of AGBO with different values of weight decay.
Weight decayMAERMSEMAPE1e-75.3026.6590.0871e-65.2476.5930.0861e-55.2666.6210.0861e-45.3006.6780.0871e-35.3636.7430.088Compared with the learning rate, the values of MAE, RMSE, and MAPE of AGBO only have slight differences and the optimal values of weight decay are 1e-6. The convergence plot of AGBO with the optimal parameters is shown in Figure10, and the loss value is converged to 0 after approximately 400 iteration steps. Compared with three SMBO algorithms, the convergence curve of AGBO is much smoother and stabler. This result can be attributed to the fact that in the gradient-based optimization method proposed in this work, the searching direction for parameter update is deterministic, while for other gradient-free methods, the searching direction is heuristic and largely dependent on the initialization, which usually leads to an oscillation in thelossfunction, making the convergence much more difficult.Figure 10
The convergence plot of RS-LR (a), TPE-LR (b), SA-LR (c), and AGBO-LR (d).
(a)(b)(c)(d)Figure 11
The bar chart with error bars from results of MAE (a), RMSE (b), and MAPE (c).
(a)(b)(c)Since, RS, TPE, SA, and AGBO all have certain degrees of randomness, each algorithm is repeated 25 times and then averaged to guarantee the correctness and stability of numerical performance, the results in three evaluation functions are shown in Table6 and Figure 11fig11. The repetition result indicates that AGBO-LR not only performs better in PDN’s parameteridentification compared with gradient-free algorithms but also has better numerical stability and robustness with the lowest variants during repeating experiments.Table 6
The results of parameter identification repeated 25 times.
AlgorithmMAERMSEMAPERS-LR6.447 ± 0.8018.054 ± 0.9580.105 ± 0.013TPE-LR6.078 ± 0.7537.589 ± 0.8300.099 ± 0.012SA-LR6.970 ± 1.1118.682 ± 1.3180.114 ± 0.018AGBO-LR5.247 ± 0.0796.593 ± 0.1110.086 ± 0.001


## 5. Conclusions

Parameter identification plays a key role in PDN calculation and analysis; therefore, some methods have been proposed to improve the accuracy of parameters in PDN, such as LS, MCMC and sequential model-based global optimization. However, most of the existing algorithms are classified as gradient-free methods. Therefore, in this work, an adaptive gradient-based optimization method is proposed for parameter identification in PDN. The analytical expressions of the gradients of the loss function with respect to the parameters are derived, and an adaptive updating scheme is utilized. We compare the proposed method with several heuristic algorithms such as RS, TPE, and SA. It is found that the errors via adaptive gradient-based methods are lower in all three evaluation functions, namely, MAE, RMSE, and MAPE, with smooth and stable convergence of the loss function. By further taking a linear transformation of the loss function, the method of this work significantly promotes the parameter identification performance with a much lower variance in 25 repeat experiments. In addition, the variations in hyperparameters of optimization methods such as learning rate and weight decay are also investigated, indicating that the method proposed in this work achieves more stable and robust performance to identify PDN parameters. It should be noted that the gradient-based optimization method can also be further explored in future work, such as the updating schedule of the learning rate for more stable and faster convergence of the loss function, the interpretability of the gradient-based method for parameter identification of PDN.



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*Source: 9300522-2022-10-15.xml*