1 Introduction

Flexible alternative current transmission system (FACTS) and high voltage direct current (HVDC) has been widely installed to boost the power transmission capacity in the AC network [1,2,3]. These devices, however, can cause problems such as torsional interaction or control interaction. The types of interactions are defined in [4] as sub-synchronous resonance (SSR), sub-synchronous torsional interaction (SSTI), and sub-synchronous control interaction (SSCI), which were experienced in the U.S, Europe, and China [5,6,7,8,9]. ENTSO-E established a network requirements code for grid connection of HVDC, which addresses the increasing concern regarding potential interactions associated with HVDC systems [10]. In Korea, thyristor-controlled series capacitor (TCSC) was installed in the Yeongdong area, and then Korea Electric Power Corporation (KEPCO) has studied the impact of SSTI in large-scale power systems with TCSC replica controller [11].

Various studies related to SSTI behavior and aspects have been reported in the literature. Several studies have shown the effect on electrical damping by changing controller parameters, short circuit level (SCL), compensation level, or power transfer level [12,13,14]. Subsynchronous damping controller (SSDC) to damp the oscillation has been proposed in [15, 16]. Reference [17] represented the method to calculate modal mechanical damping but did not include the method to calculate structural mechanical damping of the multi-mass in the turbine generator. In our previous conference paper [18], we proposed Rayleigh and Caughey damping model for calculating structural mechanical damping; however, this paper did not include the simulation results on the frequency and time domain analysis.

The structural mechanical damping can be efficiently calculated by Rayleigh and Caughey damping model [19]. These models have been mainly used in mechanical engineering, but the existing SSTI study in electrical engineering did not explain these models. We provide the method to estimate structural mechanical damping with limited information on the generator in addition to comprehensive studies in both frequency and time domain analysis. Theoretical aspects and applicability of two damping models for the SSTI analysis are also discussed and demonstrated.

The paper is structured as follows. Section 2 describes the process of SSTI investigation. Section 3 explains Rayleigh and Caughey models to calculate structural mechanical damping. Section 4 shows the results of Rayleigh and Caughey damping models. Simulation results are presented in Sect. 5. The paper is concluded in Sect. 6.

Fig. 1
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The process of SSTI study

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2 The Process of SSTI Study

Figure 1 depicts the process of SSTI study. Analysis of mechanical characteristics is the process for calculating the damping of the multi-mass of the turbine generator. Modal mechanical damping on modal analysis (Step 2) is expressed by uncoupled form with diagonal matrix for torsional modes. Structural analysis (Step 3) is the process of calculating the inherent damping of structures. Structural mechanical damping of the multi-mass can be divided into self damping, having each mass in the turbine, and mutual damping, being between masses. Analysis of electrical characteristics calculates electrical damping in the frequency domain using damping torque analysis. Electrical damping (Step 3) means damping existing in the electrical network. We assess the risk of SSTI by comparing whether the summation of the electrical and the modal mechanical damping in the frequency domain is negative or positive; in the time domain, the stability is evaluated according to whether the angular speed of the generator is damped or not. A full time domain study requires more detailed data for turbine generator, in particular the inertia, shaft stiffness, and structural mechanical damping for self and mutual damping. However, detailed data for the turbine generator is often neglected. The information in [20] provided two multi-mass data for IEEE first benchmark model (FBM) and second benchmark model (SBM). IEEE FBM assumed the self damping and mutual damping, and IEEE SBM provided self damping and modal decay rate. The structural mechanical damping replaces the amount of damping corresponding to modal mechanical damping in the frequency domain. If this damping is regarded as zero, the time domain study cannot judge the positive effect of structural mechanical damping.

3 Structural Mechanical Damping Models

The mechanical shaft system of a turbine generator unit consists of several masses, which are high pressure turbine (HPT), intermediate pressure turbine (IPT), lower pressure turbine (LPT), generator turbine (GT), etc. This system is modeled using a lumped mass-spring-damper system. The equation of motion can represent the equations for all masses [21].

$$\begin{aligned} \varvec{H\frac{d^2\theta }{dt^2}} + \varvec{D\frac{d\theta }{dt}} +\varvec{K\theta } = \varvec{T} = \varvec{T_m - T_e} \end{aligned}$$
(1)

where \(\textit{H}\) is a diagonal matrix of the inertia constants for each mass, \(\textit{D}\) and \(\textit{K}\) are tridiagonal symmetric matrices of structural mechanical damping and shaft stiffness, respectively. \(\theta\) is the angular position. \(\textit{T}\) is the applied torque from mechanical torque \((\textit{T}_m)\) and electrical torque \((\textit{T}_e)\).

Modal analysis can be performed on the transformation of (1) with the substitution of (2).

$$\begin{aligned} \varvec{\theta } = \varvec{Q\delta } \end{aligned}$$
(2)

where \(\delta\) is modal angle transformation, and \(\textit{Q}\) is a matrix whose columns consist of the set of right eigenvectors.

The mechanical system can be expressed in terms of modal parameters multiplying with \(\textit{Q}^T\) [22]. Equation (1) becomes

$$\begin{aligned} \varvec{Q^THQ\frac{d^2\delta }{dt^2}} + \varvec{Q^TDQ\frac{d\delta }{dt}} +\varvec{Q^TKQ\delta } = \varvec{Q^TT} \\ \rightarrow \varvec{H_m\frac{d^2\delta }{dt^2}} + \varvec{D_m\frac{d\delta }{dt}} +\varvec{K_m\delta } = \varvec{Q^TT} \end{aligned}$$
(3)

where \(\textit{Q}^T\) is the transpose of \(\textit{Q}\), and \(\textit{H}_m\), \(\textit{D}_m\), and \(\textit{K}_m\) indicate the modal inertia, modal mechanical damping, shaft stiffness matrices, respectively. Through decoupled form for modal analysis, \(\textit{K}_m\) becomes diagonal, and \(\textit{D}_m\) can be assumed to be a diagonal matrix.

Fig. 2
figure 2

Input parameters for the damping of multi-mass in PSCAD/EMTDC

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The reason for performing modal analysis is to define the natural frequency and modal mechanical damping of multi-mass. The most necessary information to estimate structural mechanical damping is the natural frequency of multi-mass, and modal mechanical damping is later compared with estimated modal mechanical damping. We can derive modal mechanical damping from the damping ratio for each mode. Still, we cannot directly obtain structural mechanical damping for each mass using the transpose of the inverse of a modal damping matrix in (3). Although modal mechanical damping is known, structural mechanical damping indicates an inaccurate matrix due to the influence of \(\textit{Q}\), even if the transpose of the inverse from modal to structural is performed. \(\textit{Q}\) from the result of eigen analysis did not reflect the effect of structural mechanical damping.

Modeling of damping is very complex, but the common approach is to use the proportional damping model [23]. The proportional damping model represents the damping matrix (D) as a linear combination of the mass moment of inertia (H) and shaft stiffness (K) matrices. This damping model is also known as Rayleigh damping model. Caughey damping model is a generalization of the Rayleigh damping model and expresses the series representation for inertia and shaft stiffness.

Structural mechanical damping can be included as an input parameter of the multi-mass model in PSCAD/EMTDC, as shown in Fig. 2. Then, we can verify the damping effect in the time domain. This section explains the methods to estimate structural mechanical damping through the Rayleigh and Caughy damping model.

3.1 Rayleigh Damping Model

The damping matrix known as Rayleigh damping model or proportional damping in [23] is obtained as a linear combination of mass moment of inertia and shaft stiffness, as in (4).

$$\begin{aligned}{}[{\varvec{D}}]= \alpha [{\varvec{H}}]+ \beta [{\varvec{K}}]\end{aligned}$$
(4)

where \(\alpha\) and \(\beta\) are predefined constants, called as mass and stiffness proportional coefficient respectively.

The damping ratio \((\xi _k)\) for the \(\textit{k}\)-th mode of such a system is:

$$\begin{aligned} \xi _{k} = \frac{1}{2}\left( \alpha \frac{1}{\omega _k}+\beta \omega _k \right) \end{aligned}$$
(5)

\(\alpha\) and \(\beta\) can be determined from (6), if the damping ratios of the two modes are selected.

$$\left\{\begin{array}{l}\alpha\\ \beta \end{array}\right\} = \frac{2\omega _j\omega _i}{\omega ^2_j-\omega ^2_i} \begin{bmatrix} \omega _j &{} \quad -\omega _i\\ -1/\omega _j &{} \quad {1/\omega _i} \end{bmatrix} \left\{\begin{array}{l}\xi _{i}\\ \xi _{j} \end{array}\right\}$$
(6)

where \(\omega _i \, \& \, \omega _j\) and \(\xi _i\, \& \,\xi _j\) are natural frequency and damping ratio of \(\textit{i}\)-th and \(\textit{j}\)-th modes respectively.

Rayleigh damping model properly selects two modes to ensure a reasonable damping ratio for all modes. Since the damping ratio of the unselected mode is not considered, an error inevitably occurs in other modes. However, the Rayleigh damping model is the simplest and most commonly used method for damping calculations.

3.2 Caughey Damping Model

Caughey damping model has proposed a series expression for the damping matrix regarding the mass and stiffness matrices [24]. This model can be available as \(\textit{n}>\) 3, \(\xi _1, \xi _2, \ldots , \xi _n\). The Caughey damping matrix is obtained as a linear combination of \(\textit{n}\) but involves negative or zero powers of \(\textit{H}^{-1}{} \textit{K}\). \(\alpha _k\) is an arbitrary coefficient.

$$\begin{aligned}{}[{\varvec{D}}]= \sum _{k=0}^{n-1}\alpha _{k}[{\varvec{H}}]([{\varvec{H}}]^{-1}[{\varvec{K}}])^k \end{aligned}$$
(7)

When \(\textit{i}\)-th damping ratios are specified in the \(\textit{m}\)-th degree of freedom, the damping ratio in (8) can express the \(\textit{i}\)-th natural frequency and the arbitrary coefficients.

$$\begin{aligned} \xi _{i} = \frac{1}{2}\sum _{k=0}^{m-1}\alpha _{k}\omega _{i}^{2k-1} \end{aligned}$$
(8)

The Caughey damping model can be obtained regardless of the number of modes but is numerically unstable when obtaining the power series form. In addition, when the damping ratio is specified for three or more modes, unlike inertia and stiffness matrices, the Caughey damping matrix becomes a full matrix in structures.

Fig. 3
figure 3

The flow chart of calculating the structural mechanical damping

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Table 1 Inertia and stiffness constant of multi-mass
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4 Results of Structural Mechanical Damping

IEEE FBM includes the inertia and stiffness data of the multi-mass [25]. Table 1 represents the inertia and stiffness data. A damping ratio for each mode needs to apply the described damping models in Sect. 3. However, since IEEE FBM had not included the damping ratio, we assumed that every mode has the same damping ratio, \(\xi _{0, 1, \dots , 5} = 0.0021\). Figure 3 depicts the steps to estimate the structural and modal mechanical damping, where the multi-mass model structure is shown in Fig. 4. The multi-mass from eigen and modal analysis calculates the desired modal decay rate \((\sigma _{m})\) and mechanical damping \((D_{m})\) for each mode, as shown in Table 2. In this step, structural mechanical damping is neglected. Next, the structural mechanical damping is calculated by using both damping models. The structural mechanical damping is divided into self damping \((D_{self})\) and mutual damping \((D_{mutual})\). The fourth step performs modal analysis again, considering the estimated structural mechanical damping in (3). This step to judge the accuracy of the results compare the desired modal analysis result in step 2 and the estimated modal analysis in step 4.

Fig. 4
figure 4

Multi-mass structure of IEEE FBM

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Table 2 Natural frequency, modal decay rates, and modal mechanical damping results
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4.1 Rayleigh Damping Result

The Rayleigh model calculates the structure damping by obtaining \(\alpha\) and \(\beta\) in (6). The \(\alpha\) and \(\beta\) depend on the selected two natural frequencies or modes. The multi-mass at IEEE FBM has six natural frequency \((f_{m})\) and modal decay rates, but mode 0 for nearly 0 Hz couldn’t be considered. Although this model obtains a total of 10 combinations according to the selected natural frequency, the modes can be selected in the low natural frequency because the SSTI problem generally affects the low frequency range. When modes 1 and 2 are selected, Tables 3 and 4 show the structural mechanical damping and estimated modal results, respectively. Tables 5 and 6 come from selecting modes 1 and 3.

Table 3 Structural mechanical damping estimation with mode 1 and 2
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Table 4 Estimated modal results with mode 1 and 2
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Table 5 Structural mechanical damping estimation with mode 1 and 3
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Table 6 Estimated modal results with mode 1 and 3
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It is confirmed that the estimated modal results with the selected mode in Tables 4 and 6 appear identical to the desired modal results in Table 2. The decay rate results of modes 1\(~ \& ~\)2 or 1\(~ \& ~\)3 have the same value as the desired decay rate. Rayleigh model suggests that the decay rate for the selected natural frequency is identical to the desired decay rate except for different modes.

4.2 Caughey Damping Result

The Caughey damping model in IEEE FBM has one damping matrix. Because we assumed that the damping ratio is the same, it is unnecessary to consider the number of damping ratios. The \(\alpha _k\) is calculated by (9). Substituting \(\alpha _k\) into (7), the Caughey model estimates the structural mechanical damping.

$$\begin{aligned} \begin{bmatrix} \alpha _0 \vspace{2mm} \\ \alpha _1 \vspace{2mm} \\ \alpha _2 \vspace{2mm} \\ \alpha _3 \vspace{2mm} \\ \alpha _4 \vspace{2mm} \\ \alpha _5 \vspace{2mm} \\ \end{bmatrix} = 2 \begin{bmatrix} 1/\omega _0 &{} \quad \omega _0 &{} \quad \omega _0^3 &{} \quad \omega _0^5 &{} \quad \omega _0^7 &{} \quad \omega _0^9 \vspace{2mm} \\ 1/\omega _1 &{} \quad \omega _1 &{} \quad \omega _1^3 &{} \quad \omega _1^5 &{} \quad \omega _1^7 &{} \quad \omega _1^9 \vspace{2mm} \\ 1/\omega _2 &{} \quad \omega _2 &{} \quad \omega _2^3 &{} \quad \omega _2^5 &{} \quad \omega _2^7 &{} \quad \omega _2^9 \vspace{2mm} \\ 1/\omega _3 &{} \quad \omega _3 &{} \quad \omega _3^3 &{} \quad \omega _3^5 &{} \quad \omega _3^7 &{} \quad \omega _3^9 \vspace{2mm} \\ 1/\omega _4 &{} \quad \omega _4 &{} \quad \omega _4^3 &{} \quad \omega _4^5 &{} \quad \omega _4^7 &{} \quad \omega _4^9 \vspace{2mm}\\ 1/\omega _5 &{} \quad \omega _5 &{} \quad \omega _5^3 &{} \quad \omega _5^5 &{} \quad \omega _5^7 &{} \quad \omega _5^9 \vspace{2mm} \\ \end{bmatrix}^{-1} \begin{bmatrix} \xi _0 \vspace{2mm} \\ \xi _1 \vspace{2mm} \\ \xi _2 \vspace{2mm} \\ \xi _3 \vspace{2mm} \\ \xi _4 \vspace{2mm} \\ \xi _5 \vspace{2mm} \\ \end{bmatrix} \end{aligned}$$
(9)
Table 7 Structural mechanical damping estimation using Caughey model
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Table 8 Estimated modal results using Caughey model
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Table 8 shows the estimated modal results when the Caughey model is applied, and errors occur in all modes compared with the desired modal results in Table 2. Considering every mode causes the error between the desired and estimated decay rates.

5 Simulation Results

The simulation was conducted on IEEE FBM in PSCAD/EMTDC. For the interaction study, the electrical damping for the grid was calculated using damping torque analysis in [26]. Next, Comparing the mechanical and electrical damping could confirm the impact of SSTI in the frequency domain. Finally, the multi-mass model, including the mechanical damping, had demonstrated the effect of damping in the time domain with PSCAD/EMTDC. This analysis also confirms whether the incidence of SSTI matches the frequency domain and the time domain results.

IEEE FBM in PSACD/EMTDC includes the inertia and stiffness data of the multi-mass and simulates the torsional interaction between the multi-mass in the turbine generator and series compensated transmission systems, as shown in Fig. 5. The electrical damping varies according to the compensation level of the series compensator, series capacitor. As the compensation level decreases, the electrical damping decreases, and the resonance frequency shifts to the right. The shift means the resonance frequency increase. The simulation was conducted while changing the capacitor to 21 uF, 40 uF, and 60 uF, and the electrical damping is represented in Fig. 6. The black dashed line is electrical damping when the capacitor is 21 uF with 70% compensation level, the blue is 40 uF with 40%, and the red is 60 uF with 27%. The red straight line shows the modal mechanical damping. Although the mechanical damping originally had a positive value, the sum of damping is expressed as a negative value compared to the electrical damping. Modal mechanical damping shows that the red vertical line in Fig. 6 indicates the range between full and light load damping conditions. The full load damping is the maximum absolute value, the same as Table 2. The light load damping is assumed to be 10% of full load damping.

Figure 7 is similar to Fig. 6 but put in full load damping results of the two models obtained in Sect. 4. Desired modal damping of the red star mark is the value in Table 2 derived from the damping ratio. Rayleigh of blue circle mark and Caughey of green plus mark damping indicate modal damping \((\textit{D}_m)\) of Tables 4 and 8. The probability of SSTI is very high in 70% compensate level because Rayleigh and Caughey modal damping of mode 2 are smaller than electrical damping. In the case of 40% compensation, there is the potential that interaction would occur in mode 4. Estimated modal damping for Rayleigh and Caughey models is slightly larger than electrical damping, so the sum of electrical and mechanical damping has a positive value. In 27% compensation, both damping models are less likely to cause SSTI problems because of having positive damping.

Fig. 5
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Grid configuration of IEEE FBM

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Fig. 6
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Comparison of electrical and mechanical damping of IEEE FBM changing series capacitor size

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Fig. 7
figure 7

Comparison of electrical and mechanical damping with desired, Rayleigh, and Caughey modal damping

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Fig. 8
figure 8

PSCAD simulation results with 70% compensation. a Zero structural damping in multi-mass. b Applying Rayleigh damping for Table 4 in multi-mass

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Fig. 9
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PSCAD simulation results with 40% compensation. a Zero structural damping in multi-mass. b Applying Rayleigh damping for Table 4 in multi-mass

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Fig. 10
figure 10

PSCAD simulation results with 27% compensation. a Zero structural damping in multi-mass. b Applying Rayleigh damping for Table 4 in multi-mass

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Fig. 11
figure 11

PSCAD simulation results applying Caughey damping for Table 7 in multi-mass. a 70% compensation. b 40% compensation. c 27% compensation

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Based on the frequency domain results shown in Fig. 7, the time domain analysis examines the SSTI behavior when the multi-mass structural mechanical damping has zero damping, Rayleigh model of Table 3, and Caughy model of Table 7. A three-phase to ground fault is applied at the capacitor node at t = 10 sec and is cleared after 0.075 sec. A three-phase to ground fault is considered the worst-case scenario to make oscillatory mode. Figures 8, 9, and 10 show the time domain simulation results according to compensation level. These figures divide zero damping of (a) into Rayleigh damping of (b). In the case of 70% compensation, the SSTI phenomenon should occur regardless of the structural damping because of having negative modal damping. Figure 8 represents the growing oscillation in generator speed and turbine torque among the mass. However, Fig. 9 with 40% compensation depicts that Fig. 9a has the undamped system, and Fig. 9b eventually settles down. Figure 10 also has similar results. When the sum of modal mechanical damping and electrical damping is positive value like Fig. 7 with 40% and 27% compensation, it is demonstrated that the presence of structural damping can restrict oscillation for torsional interaction mode in the time domain. Figure 11 describes the results, including the Caughey damping model. Figure 11a with 70% compensation only shows the oscillation in the generator speed and turbine torque.

In several papers, only the electrical damping was derived, and the time domain simulation was performed, ignoring the structural mechanical damping. However, in this paper, the time domain results applied with structural mechanical damping demonstrate whether SSTI occurs or not according to the sum of electrical damping and modal mechanical damping in the frequency domain.

6 Conclusion

This paper has presented the practical estimation method of structural mechanical damping with limited information by adopting Rayleigh and Caughey damping models. The Rayleigh damping model effectively provides structural mechanical damping by selecting two adjacent modes; however, the Rayleigh model is a lack of accuracy in the rest of the modes. The Caughey damping model estimates the modal decay rates considering all modes, while the Caughey model is not suitable for general use because of computational issues. The structural analysis in this paper harmoniously deploys both models, providing appropriate damping results with computational efficiency.

In the case studies, the possibility of SSTI has been assessed in the frequency and time domain. The highest compensation level case shows the torsional instability of the turbine generator with a lower modal mechanical damping value than negative electrical damping in the frequency domain. On the other hand, turbine generators in the lower level cases damp the oscillation, and the generator does not experience instability, i.e., the sum of damping is positive.

These results indicate that the structural mechanical damping provides the ability to damp oscillation and the multi-mass model in the time domain simulation needs to include this damping for accurate SSTI analysis results. Consequently, this work provides a comprehensive analysis and experimental results about the structural mechanical damping to the researchers.