## 3 Numerical Application  
The proposed scheme and model are used to compute the transient temperature fields of a prototype insulated-gate bipolar transistor (IGBT) module (Figure 2(a)) to validate its advantages. The temperature field is computed by the network topological approach [23]. Figure 2(b) shows meshes of the NTM for the prototype IGBT module. In the numerical implementation of NTM, the numbers of the total elements and nodes are, respectively, 3364 and 4303, with the number of the NTM equations equal to 4303. Obviously, the solution of such a large equation set will consume an enormous amount of computational resources. To simulate the wide-frequency-band transient response, the frequency broadband of excitation signals consisted of 100 liner-spaced samples from 10 Hz to 100 kHz. For performance comparisons, the NTM, the proposed reduced model and the conventional block-Arnoldi reduced model are used to compute the transient temperature field of this case study under the predefined, tested broadband excitation signal. In order to take both the minimum and maximum frequencies into consideration, the time step should be small enough and the total excitation time large enough. Consequently, the total excitation time was set as 0.2 s, the discrete number as 80,000 and the time step as 2.5 us. The equation was solved by ode15 s of MATLAB. In the solver settings, the relative tolerance (RelTol) was set as \(10^{-3}\) and the absolute tolerance (AbsToI) as \(10^{-6}\). The final expansion points used were: 10, \(10^{5}\), \(10^{3}\), and \(10^{2}\). As explained in the adaptive scheme, \(N_{\max}^{\text{moments}}\) is equal to the smallest number of moments under which the error between the ROM and the full model is smaller than the tolerance at the frequency. \(N_{\max}^{\text{moments}}\), at 10, \(10^{5}\), \(10^{3}\), and \(10^{2}\), is finally set as 6, 6, 7, and 9, respectively. Finally, \(N_{\max}^{\text{s}}\) is set as 100 to limit the number of total expansion points used in the MOR procedure. Moreover, the local error tolerance \(tol^{l}\) is set as 0.005, with the global error tolerance set as 0.01. Typical observing points (points 1, 2, and 3, see Figure 3b) are used for performance comparison of different ROMs. The numerical results are presented in Figures 4-9. Figures 4-6 present the time-domain response observed on points 1, 2, and 3 of the NTM model and those of the proposed reduced model (size = 28), respectively. From the numerical results, it is obvious that the proposed reduced-order system can accurately imitate the dynamic response of the original high-fidelity model. Moreover, Figures 7-9 provide the absolute errors observed on points 1, 2, and 3, between the results of the NTM model and those of the proposed reduced model (size = 28), and those of the conventional block-Arnoldi reduced model (size = 30), respectively. From the numerical results (Figures 7-9), one can see that for a dynamic response to a broadband frequency signal, the absolute transient errors of the proposed model are much smaller than those for the conventional reduced model. For point 1, the maximum error of the proposed model is 0.3603, whereas the conventional reduced model can reach 3.2275. For point 2, the maximum error of the proposed approach is 0.0986, whereas the conventional reduced model can reach 0.9803. For point 3, the maximum error of the proposed model is 0.1459, whereas the conventional reduced model can reach 6.3752. More performance comparison statistics are shown in Tables 1 and 2. Table 1 compares the transient response of different observing points using the proposed and NTM models. Table 2 compares the absolute  
Figure 3: Prototype (**a**) IGBT module and its (**b**) NTM model. errors on different observing points using the proposed and conventional schemes. The numerical results in Tables 1 and 2 also validate the conclusion obtained from Figures 4-9.