1. Introduction
Power system research mainly relies on mathematical model-based simulations. The complete simulation model of the power system fuses the models of generation, transmission, distribution, and power consumption equipment. Therefore, the accuracy of various equipment models is a prerequisite for the authenticity of power system simulation results. Since the working principles of electrical equipment are clear, the equations of these equipment models can be considered correct. Therefore, the difficulty of power system modeling is to obtain accurate model parameters. At present, parameter identification is an important way to obtain the true parameter values of the electrical equipment model. Therefore, improving the accuracy of parameter identification is of great significance for power system modeling and simulation.
Parameter identification is essentially an optimization problem that involves finding the optimal values within the parameters’ possible range of values so the model output is as consistent as possible with the measured results. The determination of the parameter value range must conform to the physical background of the parameter but also requires strong engineering judgment. It is very important to determine the identifiability of parameters and their exact value range, otherwise the identification results will have no practical meaning. The current widely used power system parameter identification process is shown in
Figure 1. There are three main steps in the process. The first step is to analyze the identifiability of the parameters to ensure that the value of the parameter can be uniquely determined in theory [
1,
2]. The identifiability can be analyzed by formula derivation [
3] or numerical methods based on time-domain sensitivity trajectories [
4]. Parameters that are not identifiable do not participate in the identification process. The second step is to analyze the possibility of the accurate identification of the identifiable parameters based on sensitivity. Generally, parameters with high sensitivity are regarded as key parameters in the model and are easy to accurately identify; parameters with low sensitivity are regarded as non-key parameters, and the identification accuracy is usually low. Since there are often many parameters in a model, when all the parameters are identified at the same time, the number of iterations of the identification algorithm needs to be significantly increased as the risk of falling into a locally optimal solution increases. Therefore, in the engineering practice of power system parameter identification, only the key parameters are identified, and the unidentifiable parameters and non-key parameters are usually set to typical values or empirical values [
5,
6,
7,
8,
9]. The third step is to identify the selected key parameters based on the dynamic response of the electrical equipment under the actual disturbance of the power grid. Parameter identification has been widely used in various electrical equipment in power systems, such as synchronous generators [
10,
11], excitation controllers [
12,
13], transmission lines [
14,
15], electric loads [
16], renewable power generation equipment [
17,
18,
19], and energy storage system [
20].
From the existing practical application of parameter identification, it can be found that sensitivity analysis plays an important role in the identification process. However, the sensitivity analysis method used in power system parameter identification is mainly trajectory sensitivity, which is a local sensitivity analysis (LSA) method that considers the impact of a single parameter change on the model output. In the family of sensitivity analysis methods, there is also a type of global sensitivity analysis (GSA) method [
21,
22,
23]. The GSA method analyzes the influence of each input or parameter on the model output when all model inputs or parameters change randomly. Compared with LSA, GSA can more comprehensively analyze the impact of the uncertainty of inputs or parameters on model output. The GSA method has a wide range of applications in many disciplines, such as hydrological modeling [
24], biomedical science [
25], earth system modeling [
26], evaluating ecological resilience [
27], building performance analysis [
28], train traffic scheduling [
29], and wind resource assessment [
30]. In recent years, the application of GSA methods in power systems has gradually increased, including the analysis of the uncertainty of renewable energy generation [
31,
32] and its relationship with the reliability of the power system [
33], the voltage control of the distribution network [
34] and the voltage stability of the transmission grid [
35,
36,
37], the analysis of the maximum load ability of islanded microgrids with distributed generation [
38,
39], the analysis of the key influencing factors of small disturbance stability [
40,
41], the analysis of the influence of various parameters on locational marginal prices in electricity market [
42], the analysis and optimization of key parameters of power generation system [
43,
44], the improvement of the power output estimation model [
45], and the cost model [
46] of wind power generation. In these studies, the variance-based Sobol method was the main GSA method used. However, there are few studies on the use of GSA in power system parameter identification. Reference [
47] applies the Sobol method to the parameter identification of the load model. By only identifying the high-sensitivity parameters, the identification difficulty is reduced, and the identification efficiency is improved. Reference [
48] also used the Sobol method to analyze the sensitivity of seven parameters in the electrical model of the lithium-ion battery, divided the parameters into three groups according to the sensitivity, and proposed a groupwise alternating identification strategy (GAIS) to iteratively identify the three groups of parameters, achieving a good fit with the experimental data. Compared with the existing method of only identifying high-sensitivity parameters, although the GAIS is more complicated in steps, it has the advantage of not only increasing the number of identification parameters but also improving the identification accuracy, so we believe that this method has a good application future. We used the GAIS in
Section 6 and made some improvements to it.
Overall, the use of LSA in power system parameter identification has a long history, and in recent years, GSA has gradually begun to be adopted. However, there is no clear conclusion on the impact of different sensitivity analysis methods on parameter identification results. This paper compares and studies the roles that LSA and GSA can play in different parameter identification methods, providing clear guidance for the selection of sensitivity analysis methods and parameter identification methods. The LSA method used in this paper is the widely used trajectory sensitivity analysis, and the GSA methods used include the Sobol method, Morris method, regional sensitivity analysis, scatter plots, and Andres visualization test. In our research, the Sobol method was implemented by programming, and the other four GSA methods were implemented using an open-source GSA toolbox named SAFE [
49,
50]. All these sensitivity analysis methods are introduced in
Section 2. A generator excitation system model is introduced as a parameter identification object in
Section 3. The reason for choosing this model is that the identifiability of its parameters can be analytically analyzed by formula derivation, which is impossible for other complex models. In
Section 4, we analyzed the sensitivity of excitation system model parameters using the LSA and GSA methods and compared the differences in the analysis results. In
Section 5 and
Section 6, we used the traditional parameter identification method shown in
Figure 1 and the improved GAIP method proposed in
Section 6 for parameter identification, respectively. Following that, we compared and analyzed the selection of sensitivity analysis methods and parameter identification methods from the identification accuracy and other perspectives, and finally provided clear recommendations.
3. Model of Excitation System
There are many types of synchronous generator excitation system models. We took the excitation system model shown in
Figure 2 as an example to compare the LSA method and the GSA methods. This model is called Type-I AVR Model in a widely used power system simulation software PSASP V7.35 in China and can be found in the Dynamic Element Model Library User’s Manual of this software. For the convenience of analysis, the saturation elements in the model have been omitted, and the element for calculating the voltage deviation Δ
U has been moved before the filter. The input of the model is −Δ
U, and the output of the model is the variety of excitation voltages Δ
E. There are seven parameters in this model, as listed in
Table 2.
It should be noted that the excitation system model is used as an example not because obtaining the model parameters is difficult in engineering, as many of them can be obtained from the OEM datasheet, but because the identifiability of all parameters of this model can be analytically analyzed by the following formula derivation. Thus, when we discuss the causes of large errors in parameter identification below, we can clearly point out the identification error of which parameter comes from the identifiability problem and that of which parameter comes from the identification method problem. The identification method can be improved; however, the identifiability problem is determined by the model structure and input/output variables. Even if the input/output variables can be changed, they may not be able to solve the identifiability problem.
The identifiability of all parameters in this model is derived as follows:
According to the block diagram shown in
Figure 2, the complete transfer function
G(
s) of the excitation system can be obtained as Equation (7). The expressions of each coefficient in
G(
s) are shown in Equations (8)–(13).
For
G(
s) in Equation (7), according to the measured data of Δ
U and Δ
E, the coefficients {
a4,
a3,
a2,
a1,
b1,
b0} can be identified first. Following that, the seven parameters can be solved according to Equations (8)–(13). The identifiability of the seven parameters listed in
Table 2 is analyzed through formula derivation as follows.
First, we found that KaKr appears only in Equations (12) and (13) and that KaKr as a whole can be directly obtained from Equation (13), resulting in values of Ka and Kr that cannot be uniquely determined. Therefore, KaKr is identifiable, while neither Ka nor Kr are identifiable.
Then, according to Equations (12) and (13),
Tf can be obtained as
Substituting Equation (14) into Equations (8) and (11), we obtain
Substituting Equation (15) into Equations (9) and (10), we obtain
According to Equations (16)–(18), the values of
Tr, (
Ta +
Te), and
KfKa can be determined as
In Equation (19), the equation of
Tr is a one-variable quaternary equation whose root-finding formula is too complicated to use to analyze the results. Therefore, we adopted numerical analysis. According to the typical values of the parameters listed in
Table 2, we can obtain the four roots of
Tr as
Considering that both (
Ta +
Te) and
KfKa should be greater than zero,
Tr(1) can be excluded. Considering that the value of
Tr is usually in the range of [0.00 s, 0.06 s],
Tr(2) is excluded. However,
Tr(3) and
Tr(4) cannot be further screened. Therefore, we obtained two groups of solutions for Equation (19) as follows:
Considering that the difference between the two solutions in Equation (21) is less than 1%, it is approximately considered that Tr, (Ta + Te), and KfKa are identifiable.
Finally, after the value of (Ta + Te) is obtained, the values of Ta and Te can be obtained according to Equation (15). Therefore, both Ta and Te are identifiable. Although the value of KfKa can be obtained, KfKa always appears as a whole. Therefore, KfKa is identifiable; however, neither Kf nor Ka are identifiable.
In summary, in the excitation system model, the parameters
KaKr,
KfKa,
Tr,
Ta,
Tf, and
Te are identifiable, while
Kr,
Ka, and
Kf are unidentifiable. Therefore, we want to emphasize that the high sensitivity of parameters does not mean that parameters can be uniquely identified. The sensitivity analysis results in the next section show that the sensitivities of
Kr,
Ka, and
Kf are significantly higher than other parameters; however, their values cannot be uniquely determined according to the identifiability analysis result. In
Section 5 and
Section 6, both
KaKr and
KfKa are taken as single parameters for identification.
4. Sensitivity Analysis Results
This section will use the LSA method introduced in
Section 2.1 and five GSA methods introduced in
Section 2.2 to perform sensitivity analysis on the excitation system model parameters introduced in
Section 3 and compare the analysis results.
The input used in the following sensitivity analysis is a voltage sag with a duration of 0.2 s and depth of 0.1 p.u.
For GSA, the output of the model is defined as the mean relative error (MRE) between the response of the excitation system with unchanged parameters and the response after parameter changes as follows:
where
N is the number of data points on the excitation system output curve;
k is the index of the data point;
y0 is the response with unchanged parameters;
yc is the response after parameter changes; both
y0 and
yc are the sum of the steady-state value of the excitation voltage and the variety of excitation voltage output by the excitation system model.
4.1. LSA Results
To facilitate the comparison of the sensitivity of each parameter, we normalized the LSA results based on the maximum sensitivity value.
Figure 3 shows the normalized values of the local sensitivity of the seven parameters.
There is no uniform standard for determining the key parameters that need to be identified according to the sensitivity, and it is often determined based on the experience of the researcher. We took 1/10 of the maximum sensitivity as the boundary for distinguishing high and low sensitivity. Therefore, the most sensitive parameters Kr and Ka are definitely recognized as key parameters. Since the sensitivity of Kf, Tf, and Te is larger than 1/10 of the maximum sensitivity, they are also regarded as key parameters. The sensitivities of Tr and Ta are only approximately 1/15 and 1/16 of the maximum sensitivity, respectively, and they can be regarded as non-key parameters.
4.2. GSA Results
We refer to the typical values of the excitation system parameters in a simulation software named PSASP V7.35 which is developed by China Electric Power Research Institute and set the value ranges of the seven parameters as listed in
Table 3. When performing GSA, each parameter is set to be uniformly distributed within its value range, and Latin hypercube sampling is used when sampling the parameters, which is a commonly used GSA setting. To ensure the convergence of the analysis results of the numerical GSA methods, the number of parameter samples is 5000. For the graphical GSA methods, the number of parameter samples is 3000, and the generated graphics are clear enough to judge the sensitivity of the parameters.
4.2.1. Sobol Method
Figure 4 shows the normalized sensitivity values of the excitation system parameters obtained using the Sobol method. Although the ranking of parameter sensitivity is consistent with the LSA result, the sensitivity difference between the high- and low-sensitivity parameters is significantly greater than the LSA result. For either the first-order sensitivity or the total-order sensitivity, the sensitivity of the three gain parameters
Ka,
Kr, and
Kf is much greater than the sensitivity of the four time constants
Tr,
Ta,
Tf, and
Te; therefore, the three gain parameters can be selected as the key parameters that need to be identified. The difference between the sensitivity of the four time constants and the maximum sensitivity is far greater than 10 times, so they can be considered non-key parameters.
4.2.2. Morris Method
Figure 5 shows the normalized sensitivity values of the excitation system parameters obtained using the Morris method. The sensitivity difference between the high- and low-sensitivity parameters is also several times the result of the LSA method. The ranking of parameter sensitivity is consistent with the results of both the LSA and Sobol methods. In addition to the three gain parameters that should be regarded as key parameters, the difference between the average value and standard deviation of the elementary effects of
Te and the corresponding maximum value is slightly less than 10 times, so
Te can also be regarded as a key parameter.
4.2.3. RSA Method
The analysis result of RSA is shown in
Figure 6. According to the RSA analysis method described in
Section 3, we also divided the model output into 10 groups to obtain 10 CDF curves for each parameter.
Figure 6 shows that the 10 curves of
Kr,
Ka, and
Kf are scattered, especially the curves of
Kr and
Ka, which represent their high sensitivity. Among the four time constants, the CDF curves of
Tr are relatively scattered, but the degree of dispersion is not as good as the three gain parameters, and the CDF curves of other time constants are very close. Therefore, only three gain parameters can be considered as key parameters from the graph, which is consistent with the results of the Sobol method.
4.2.4. Scatter Plot Method
Figure 7 shows the one-dimensional scatter plot of each parameter of the excitation system. According to the scatter plot interpretation method described in
Section 3, the parameters with high sensitivity are still three gain parameters because the relationship between the parameter value and the reduction in the MRE can be clearly seen. Because the scatter plots of the four time constants are very scattered and there is no clear relationship between the parameter values and the MRE value, their sensitivity is low. The above conclusion is consistent with the conclusion derived from the RSA graph.
Two-dimensional scatter plots can be used to reveal the relationship between two parameters.
Figure 8 shows the two-dimensional scatter plot of the three gain parameters, where yellow spots can be observed. Taking the two-dimensional scatter plot of
Kr–
Ka as an example, according to the position of the yellow spot, it can be concluded that the MRE only increases when the values of
Kr and
Ka approach the upper limit of their value range at the same time. Therefore, there is a correlation between
Kr and
Ka. Similarly, the correlation between
Ka–
Kf and
Kr–
Kf can be analyzed. Because no correlations are found in the two-dimensional scatter plots of other parameters, those graphs are omitted.
4.2.5. AVT Method
Figure 9 shows the AVT results of all parameters of the excitation system. The graphs of the three gain parameters in the (Y, Y1) scatter plot are obviously asymmetric to the diagonal; in the (Y, Y1) scatter plot, their graphs are not horizontally distributed. In the (Y, Y1) scatter plots of four time constants, the graphs are close to the diagonal; in the (Y, Y2) scatter plot, the graphs of
Tr,
Ta, and
Tf are basically horizontal lines. According to the interpretation method of the AVT results, the sensitivity of the three gain parameters is much greater than that of the four time constants. The sensitivity of
Te is slightly greater than that of the other three time constants.
4.3. Comparison of the LSA and GSA Results
Comparing the use of LSA and GSA, as well as the above analysis results, we can obtain the following conclusions:
In terms of the amount of calculation, the GSA is far more than that of LSA. The relationship between the number of times that various sensitivity analysis methods calculate the model output, the number of parameters
N, and the number of parameter samples
Ns is summarized in
Table 4. If the single calculation of the model output is time-consuming, unless a suitable algorithm is found [
52,
53], the analysis speed of the GSA method may be unacceptable.
Both LSA and GSA can be used to distinguish between key and non-key parameters. Although the LSA method and the numerical GSA method have the same parameter-sensitive ordering, the key parameters determined by the two kinds of methods are different. The key parameters in the LSA result are {Kr, Ka, Te, Kf, Tf}. When integrating the results of the five GSA methods, the key parameter is {Kr, Ka, Kf}. The reason is that the difference in parameter sensitivity is more significant in the GSA results, resulting in fewer key parameters being found.
Although the analysis process and result display form of the five GSA methods are different, the conclusions are the same. Therefore, there is no need to use multiple GSA methods at the same time. Because the results of numerical methods are clearer and can be used to rank parameter sensitivity, we recommend the numerical GSA method.
We compared the differences in the use and analysis results of LSA and GSA in this section. However, it is difficult to evaluate which method better achieves parameter identification from these two aspects only. In the next two sections, we used different parameter identification strategies to analyze the impact of different key parameter combinations obtained by GSA and LSA on the identification accuracy.
5. Comparison under Existing Parameter Identification Strategy
This section will use the commonly used parameter identification methods to identify the parameters of the excitation system and discuss the effectiveness of various sensitivity analysis methods according to the accuracy of parameter identification.
According to the existing parameter identification process shown in
Figure 1, the key parameters are identified, and the non-key parameters take typical values or empirical values. Since the value of the non-key parameters of the actual equipment is not clear, it is difficult to determine how accurate the typical value or the empirical value is. Therefore, we randomly selected non-key parameters within their value ranges and then analyzed the identification results of key parameters.
According to the identifiability analysis result in
Section 3, both the
KaKr and
KfKa should be identified as a whole, and their accurate values are 20.00 and 0.800, respectively.
A PSO algorithm with linearly decreasing weight coefficients [
54] was adopted for parameter identification. The number of particles is 20, the number of iterations is 200, the learning factors C1 and C2 are set to 2, and the weight factor decreases linearly from 0.9 to 0.4. The fitting error index uses the MRE in Equation (22).
5.1. Identification Result According to LSA
According to the LSA results, the key parameters to be identified are
KaKr,
KfKa,
Te, and
Tf. The non-key parameters
Tr and
Ta take random values, and their different values have a significant impact on the identification results.
Table 5 lists the results of 10 identifications.
Emin,
Emax, and
Eavr in
Table 5 refer to the minimum, maximum, and average values of the identification errors, respectively.
From the identification results, only the combination of the two highest sensitive parameters KaKr has good identification accuracy, while the identification accuracy of other parameters is not good because their average identification error exceeds 10%. Therefore, although the sensitivity of the non-key parameters Tr and Ta is very small, their imprecision has a significant negative impact on the identification accuracy of the key parameters. Although the key parameters in the model can be found through sensitivity analysis, identifying only the key parameters cannot ensure the improvement of identification accuracy. The accuracy of non-key parameters also plays an important role in improving the accuracy of parameter identification.
5.2. Identification Result According to GSA
Combining the results of the five GSA methods, we set the key parameters that need to be identified as
KaKr and
KfKa, while
Tr,
Ta,
Tf, and
Te take random values.
Table 6 lists the results of 10 identifications. A comparison of the results of
Table 5 and
Table 6 indicates that since a total of four non-key parameters do not participate in the identification according to the GSA results, the identification accuracy of the two key parameters
KaKr and
KfKa is significantly lower than that of the LSA-result-based identification. This result again shows that the accuracy of low-sensitivity parameters cannot be ignored in parameter identification.
5.3. Example of High Sensitivity Not Equating to Identifiability
The derivation of the formula in
Section 3 has shown that the single parameters
Kr,
Ka, and
Kf are not uniquely identifiable, and only their products
KaKr and
KfKa are uniquely identifiable.
Table 7 shows the identification results of
Kr,
Ka, and
Kf corresponding to the
KaKr and
KfKa in
Table 5. As can be seen from
Table 7, since the errors of
KaKr and
KfKa are small, the fitting error of the model output is already small, but the identification results of the three gain parameters
Kr,
Ka, and
Kf are still very scattered. Small model fitting errors but scattered parameter identification results are typical features of identifiability problems, which also proves the correctness of the derivation in
Section 3. According to the sensitivity analysis results in
Section 4, the sensitivity of parameters
Kr,
Ka, and
Kf is significantly greater than that of other parameters. This example clearly shows that the high sensitivity of a parameter is not equivalent to a parameter being identifiable.
5.4. Discussion of the LSA-Based and GSA-Based Identification Results
By comparing the LSA-based and GSA-based parameter identification results, we found that the small sensitivity of the parameter means that it has only a small impact on the model output. This does not mean that its deviation from the true value has little effect on the identification accuracy of other parameters. Without knowing the exact value of the non-key parameter, when more non-key parameters do not participate in the identification, the possibility of the key parameters being accurately identified decreases. Because GSA amplifies the sensitivity difference between the high- and low-sensitivity parameters, the non-key parameters that are not involved in the identification increase, resulting in a decrease in the identification accuracy of the key parameters.
In general, under the existing identification strategy that identifies only key parameters, GSA does not show advantages over LSA. In the next section, to solve the negative impact of inaccurate non-key parameters on the identification accuracy, a modified groupwise alternating identification method of high- and low-sensitivity parameters was used to compare the effects of LSA and GSA.
7. Conclusions
Sensitivity analysis plays an important role in the parameter identification of power systems. The use of LSA has a long history, and in recent years, GSA has gradually begun to be adopted. However, there is no clear conclusion on the impact of different sensitivity analysis methods on parameter identification results. Therefore, this paper compares and studies the roles that LSA and GSA can play in different parameter identification methods, providing clear guidance for the selection of sensitivity analysis methods and parameter identification methods. The conclusions are as follows:
The calculation amount of the GSA methods is much larger than that of the LSA method, especially the numerical GSA methods. The GSA method may be inconvenient to use in a model that takes a long time for a single calculation;
The results of the five GSA methods on the grouping of high- and low-sensitivity parameters are the same. Because the difference in the high- and low-sensitivity values is more prominent in the GSA results, the grouping results of the key and non-key parameters are different from the LSA method;
Under the strategy of identifying only key parameters, the identification accuracy based on the GSA is not as good as that based on the LSA when the non-key parameters are inaccurate because the GSA enlarges the difference between high- and low-sensitivity values, resulting in more non-key parameters found;
When the groupwise alternating identification strategy of high- and low-sensitivity parameters is used, the identification accuracy based on the LSA or GSA is equivalent. However, LSA is better than GSA in terms of the corresponding relationship between identification accuracy and sensitivity values.
In summary, for the example used in this paper, both the GSA and LSA can be used to find the key parameters in the model; however, the GSA methods do not show absolute advantages over the LSA method. To improve the identification accuracy, it is more important to improve the identification strategy than to change the sensitivity analysis method. If the identification strategy that identifies only key parameters is adopted, we still recommend using the existing LSA method. If the GAIS of high- and low-sensitivity parameters is adopted, either LSA or GSA can be used. In addition, through the example used, we also want to emphasize that the high sensitivity of a parameter does not prove that this parameter must be identifiable.