1. Introduction
Physical simulation technology is often used to investigate steady-state characteristic and transient-state characteristics of transmission lines [1]-[4]. Even though real-time digital simulation technology is widely used [5], physical simulation technology still holds an important position because it can intuitively reflect changes in electricity, especially under high voltage conditions. Generally, \(\pi\)-type and T-type circuits, or occasionally \(\Gamma\)-type circuits are applied to physically simulate the transmission line [6]-[9]. These three lumped parameter models (LPMs) have high analog accuracy in systems with low frequency and short lines. Due to the low operating frequency and short transmission distance, the wavelength of the voltage is so small that it can be ignored. For the network with high frequency and long lines [10], [11], such as high voltage direct current and half-wavelength AC transmission, the wavelength of the voltage accounts for a large proportion of the length of the transmission line [12]-[14]. Equivalent LPMs of these systems have significant errors. In these systems, transmission lines have been integrated with series impedance and parallel admittance, which is known as distributed parameter effect [15]-[19].
Line resonance is caused by the interaction of parasitic impedance and capacitance to ground [20], [21]. Distributed parameter effect leads to inconsistent resonance characteristics between the line and the equivalent line. Therefore, the resonance characteristics of the LPMs also characterize the simulation accuracy. The distributed parameter model (DPM) has been proven to have periodic parallel and series resonance frequencies [22]. However, there is still a lack of system-level research on the impedance of multi-cascade transmission lines.
To accurately simulate the impedance and transmission characteristics of long lines, multiple-cascade short lines are often used to equivalent long lines [23]. The cascaded number not only affects the simulation accuracy but also affects the experimental cost. Too large cascades have high accuracy but also high cost. Reference [24] provided a simple method for determining the cascaded number. When the line length is less than one-thirtieth of the wavelength of the simulated highest frequency, the distributed parameter effect of the line can be ignored. In this case, \(\pi\)-type and T-type LPMs can be used to simulate frequencies below the highest frequency. However, this is still a considerable number. For example, for the 750 kV/500 km lossless line in [16], 155 segment LPMs are required to simulate the transmission characteristics of 3000 Hz. Reference [25] proposed a method for determining the cascaded numbers of cascaded \(\pi\)-type and T-type LPMs of lossless lines based on mathematical analysis. References [26], [27] discussed the difference in transient-state characteristics between DPM and LPM, and further determine the cascaded number through simulation. But the implementation is complex.
This paper discusses the difference in the resonance characteristics between DPM and LPMs of transmission lines. A method for determining the cascaded number for different cascaded LPMs is further presented. Based on resonance mode analysis (RMA) and theoretical analysis, the periodic resonance characteristics of DPM in open and short circuits are studied. The resonance differences between DPM and LPMs are discussed. The characteristic impedance of cascaded \(\pi\)-type and T-type LPM is derived from the single-segment LPM. The consistency of first resonance frequencies indicates that cascaded \(\pi\)-type and T-type LPMs have the same accuracy. The method for determining cascaded number based on the first resonance frequency of the single segment shows that the cascaded \(\Gamma\)-type LPM has low accuracy. Finally, the proposed determination method for LPMs is verified by simulation.
2. Resonance Characteristics of Transmission Lines
2.1 Resonance Mode Analysis Method
Harmonic resonance means a drastic energy exchange process in the network. The RMA technique can find the impedance elements with violent energy exchange in the resonance network [28], [29]. It is based on the eigenvalue decomposition of the admittance matrix and the implementation is expressed as follows:
| \[\begin{align} \boldsymbol{V}_{f} = \boldsymbol{Y}_{f}^{-1} \boldsymbol{I}_{f} \tag{1} \end{align}\] |
where \(\boldsymbol{V}_{f}\) is the node voltage matrix. \(\boldsymbol{Y}_{f}\) is the node admittance matrix. \(\boldsymbol{I}_{f}\) is the nodal current injection matrix. The eigenvalue decomposition of \(\boldsymbol{Y}_{f}\) is expressed as
| \[\begin{align} \boldsymbol{Y}_{f} = \boldsymbol{L} \boldsymbol{\varLambda} \boldsymbol{T} \tag{2} \end{align}\] |
where \(\boldsymbol{L}\) and \(\boldsymbol{T}\) are the left eigenvector matrix and right eigenvector matrix, respectively. \(\boldsymbol{\varLambda}\) is the diagonal eigenvalue matrix. Substituting (2) into (1) yields
| \[\begin{align} \boldsymbol{V}_{f} = \boldsymbol{L} \boldsymbol{\varLambda}^{-1} \boldsymbol{Y}_{f}^{-1} \boldsymbol{L}^{-1} \boldsymbol{I} \tag{3} \end{align}\] |
\(\boldsymbol{U}_{f} = \boldsymbol{L}^{-1} \boldsymbol{V}\) and \(\boldsymbol{J}_{f} = \boldsymbol{L}^{-1}\boldsymbol{I}\) are defined as modal voltage vector and modal current vector, respectively. Equation (3) can be expressed as
| \[\begin{align} \begin{bmatrix} U_{f1} \\ U_{f2} \\ \vdots \\ U_{fn} \\ \end{bmatrix} = \begin{bmatrix} \lambda_{f1}^{- 1} & 0 & 0 & 0 \\ 0 & \lambda_{f2}^{- 1} & 0 & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & \lambda_{fn}^{- 1} \\ \end{bmatrix} \begin{bmatrix} J_{f1} \\ J_{f2} \\ \vdots \\ J_{fn} \\ \end{bmatrix} \tag{4} \end{align}\] |
The reciprocal of the \(\lambda_{fi}\) is resistive. The modal eigenvalues of resonance can be determined according to the eigenvalues. Therefore, RMA can quickly locate the resonance frequency and harmonic center.
2.2 Resonance Characteristics of DPM
When the signal transmitted on the line is at power frequency, the length is much smaller than the signal wavelength, and the line is considered a short line. When the transmitted signal is in the high-frequency range, the length is in the same order of magnitude as the signal wavelength. The analysis cannot be based on the circuit theory at low frequencies. The resonance characteristics should be analyzed using the equivalent circuit with distributed resistance, capacitance, and inductance. Figure 1 shows the distributed parameter model of the line. In the figure, \(R_{0}\), \(L_{0}\), \(G_{0}\ (=0)\), \(C_{0}\) are the inductance, resistance, conductivity, and susceptance of unit length. The parameters of the DPM are shown in Table 1. According to transmission line theory, the input impedance \(Z_{in}\) of the line is
| \[\begin{align} Z_{in} = Z_{c}\frac{Z_{l} + Z_{c}{\tanh}(\gamma l)} {Z_{l}{\tanh}(\gamma l) + Z_{c}} \tag{5} \end{align}\] |
where \(Z_{l}\) is the load impedance at the end of the line. \(Z_{c}\), and \(\gamma\) are characteristic impedance, and line propagation coefficient. They are expressed as
| \[\begin{align} \begin{cases} Z_{c} = \sqrt{{(R_{0} + j\omega L_{0})}/{j\omega C_{0}}} \\ \gamma = \sqrt{j\omega C_{0}(R_{0} + j\omega L)} \\ \end{cases} \tag{6} \end{align}\] |
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Fig. 1 The DPM of the transmission line. |
Table 1 Transmission line parameters. |
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When the transmission line is open, \(Z_{in}\) is
| \[\begin{align} Z_{in} = - jZ_{c} \cot(\omega l \sqrt{L_{0}C_{0}}) \tag{7} \end{align}\] |
The amplitude-frequency characteristics of \(Z_{in}\) is shown in Fig. 2. Figure 2 shows that the input impedance alternately exhibits inductance and capacitance as the frequency increases. The series resonance frequency and parallel resonance frequency are periodic, and the periods are the same. According to (7), the first series and parallel resonance frequencies are
| \[\begin{align} f_{fs} = 1 \big/ (4l\sqrt{L_{0}C_{0}}) = \mbox{395\,Hz},\quad f_{fp} = 2f_{fs} \tag{8} \end{align}\] |
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Fig. 2 The impedance characteristics of the input impedance. |
When the line end is shorted, the input impedance \(Z_{in}\) of the transmission line is
| \[\begin{align} Z_{in} = Z_{c} \coth (\gamma l) = j Z_{c} \tan(\omega l\sqrt{L_{0}C_{0}}) \tag{9} \end{align}\] |
The amplitude-frequency characteristics of input impedance is shown in Fig. 2. It can be seen that the input impedance also alternately presents inductance and capacitance. Similarly, it can be obtained that the first series and parallel resonance frequencies are 789 Hz and 395 Hz, respectively.
When the transmission line is open circuit and shorted circuit, the modal information is shown in Fig. 3. The resonance characteristics of the lines can be more directly understood through RMA. Figure 3 indicates that the parallel resonance frequency of the open circuit is equal to the series resonance frequency of the short circuit. Similarly, the series resonance frequency of the open circuit is equal to the parallel resonance frequency of the short circuit. This can also be verified by (8).
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Fig. 3 The modal impedance of the input impedance. |
In addition, modern transmission harmonics have the characteristics of high frequency and wide frequency domain, so the high-frequency skin effect (HFSE) of lines must be considered [30], [31]. With the HFSE, parasitic resistance increases with the harmonic frequency. The unit equivalent resistance \(r\) is
| \[\begin{align} r = \frac{R_{0}} {\displaystyle 1 - (1 - \sqrt{ {j 2 \pi f_{0}}/{s}})^{2}} \tag{10} \end{align}\] |
The resonance frequency and modal impedance of the transmission line are shown in Table 2. As can be seen, HFSE only plays a role in suppressing resonance peak. When the cascaded number is determined only by the resonance frequency, the HFSE will not affect the results.
Table 2 Modal information of the transmission line. |
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2.3 Resonance Characteristics of Cascaded \(\pi\)-Type LPM
When the end of the cascaded \(\pi\)-type LPM is shorted, the terminal capacitor will be shorted. Therefore, this paper discusses the resonance characteristic of the \(\pi\)-type LPM when the end is open. The \(\pi\)-type LPM with cascaded number \(m\) is shown in Fig. 4 (a). The corresponding admittance matrix \(\boldsymbol{Y}_{\pi \_m}\) is
| \[\begin{align} \boldsymbol{Y}_{\pi\_ m} = \begin{bmatrix} Y_{11} & Y_{12} & \cdots & 0 \\ Y_{21} & Y_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & Y_{mm} \\ \end{bmatrix} \tag{11} \end{align}\] |
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Fig. 4 The topology of cascaded LPM. |
The non-zero elements in the matrix are
| \[\begin{align} Y_{ij} = \left\{ \begin{array}{@{}l@{}r@{}} {-m}/{(R_{0}l + j\omega L_{0}l)} &\kern-50pt i = j \pm 1 \\ {(G_{0} + j\omega C_{0})l}/{m} + {2m}/{(R_{0}l + j\omega L_{0}l)} \\ &{\kern-50pt i = j = (2, \ldots, n - 1)} \\ {(G_{0} + j\omega C_{0})l}/{2m} + {m}/{(R_{0}l + j\omega L_{0}l)} \\ &{\kern-50pt i = j = 1,n}\\ \end{array} \right. \tag{12} \end{align}\] |
The modal information of the \(\pi\)-type LPM with different cascaded numbers is shown in Fig. 5. Figure 5 shows that the number of the parallel resonance frequency is the same as the cascaded number. The first resonance frequency increases with the cascaded number and gradually closes to \(f_{fp}\). In other words, the accuracy of the cascaded \(\pi\)-type LPM increases with the cascaded number. This is consistent with empirical facts.
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Fig. 5 Modal information of the cascaded LPM. |
The input impedance \(Z_{in\_\pi}\) of a single segment is:
| \[\begin{align} Z_{in\_\pi} = \frac{\displaystyle 1 - \omega^{2}\left(\frac{l}{m}L_{0}\right) \left(\frac{l}{2m}C_{0}\right)} {\displaystyle j\omega\frac{l}{2m}C_{0} \left(2 - \omega^{2}\left(\frac{l}{m}L_{0}\right) \left(\frac{l}{2m}C_{0}\right) \right)} \tag{13} \end{align}\] |
When the denominator in (13) is zero, the input impedance of each segment is the largest. The frequency at this point is one of the parallel resonance frequencies, and this frequency is
| \[\begin{align} f_{\pi\_ p} = \frac{m}{\pi l\sqrt{L_{0}C_{0}}} \tag{14} \end{align}\] |
When \(m\) is 1, 3, and 5, the \(f_{\pi\_p}\) are 502 Hz, 1506 Hz, and 2510 Hz, respectively. This can be clearly seen from the modal information diagram. Notably, the modal information diagram indicates that these frequencies are also the last resonance frequencies of the cascaded \(\pi\)-type LPM.
2.4 Resonance Characteristics of Cascaded T-Type LPM
When the end of the cascaded T-type LPM is open, the terminal impedance will be open-circuited. Therefore, this paper discusses the resonance characteristics of the T-type equivalent circuit when the end is shorted. The cascaded T-type LPM of the line with cascaded number \(m\) is shown in Fig. 4. The modal information of the cascaded T-type LPM with different cascaded numbers is shown in Fig. 5. Figure 5 (b) shows that the number of parallel resonance frequencies is the same as the cascaded number. As the cascaded number increases, the first resonance frequency of cascaded T-type LPM gradually approaches that of DPM. Therefore, the optimal cascaded number can be determined based on the first resonance frequency. This is consistent with the cascaded \(\pi\)-type LPM.
The input impedance \(Z_{in\_t}\) of a single segment is:
| \[\begin{align} Z_{in\_ t} = \frac{\displaystyle j\omega\frac{l}{2m}L_{0} \left(2 - \omega^{2}\left(\frac{l}{2m}L_{0}\right)\left(\frac{l}{m}C_{0}\right)\right)} {\displaystyle \left(1 - \omega^{2} \left(\frac{l}{2m}L_{0}\right)\left(\frac{l}{m}C_{0}\right)\right)} \tag{15} \end{align}\] |
The parallel resonance frequency is
| \[\begin{align} f_{T\_ p} = \frac{\sqrt{2}m} {2\pi l\sqrt{L_{0}C_{0}}} \tag{16} \end{align}\] |
When \(m\) is 1, 3, and 5, the \(f_{T\_p}\) calculated by (16) are 355 Hz, 1065 Hz, and 1775 Hz, respectively. Compared with Fig. 5, these resonance frequencies are not the final resonance frequencies but appear in the middle. This is different from the cascaded \(\pi\)-type LPM.
2.5 Resonance Characteristics of Cascaded \(\Gamma\)-type LPM
When the line is open circuit and short circuit, the cascaded \(\Gamma\)-type models of the line with cascaded number \(m\) are shown in Fig. 6. The modal information of the \(\Gamma\)-type LPMs with different cascaded numbers is shown in Fig. 7. As can be seen, although both \(\Gamma\)-type models have \(2m\) energy storage elements, they have different resonance characteristics. Firstly, they have opposite phase frequency characteristics. In the low-frequency band, open circuit and shorted circuit are capacitive and inductive, respectively. Secondly, they have different resonance frequency quantities. The number of resonance frequencies in the open circuit is equal to the cascaded number minus 1. The number of resonance frequencies in the shorted circuit is equal to the cascaded number. Compared with Fig. 5 (b), it can be seen that when the line is shorted, the resonance frequency of each cascaded number is smaller than the resonance frequency of the cascaded T-type LPM. This indicates that the analog accuracy of cascaded \(\Gamma\)-type LPM is lower than that of cascaded T-type LPM with the same cascaded number. Therefore, cascaded \(\Gamma\)-type LPM requires more cascaded numbers to satisfy accuracy demand. The input impedance \(Z_{in\_o}\) of the single segment is
| \[\begin{align} Z_{in\_o} = \left(1 - \omega^{2} \frac{l}{m} L_{0} \frac{l}{m} C_{0} \right) \bigg/ \left( j\omega\frac{l}{m}C_{0} \right) = 1/Z_{in\_ s} \tag{17} \end{align}\] |
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Fig. 6 The topology of cascaded \(\Gamma\)-type LPM. |
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Fig. 7 Modal information of the \(\Gamma\)-type LPMs. |
The corresponding series resonance frequency \(f_{s}\) and parallel resonance frequency \(f_{p}\) are both
| \[\begin{align} f_{s} = f_{p} = \frac{m}{2\pi l\sqrt{L_{0}C_{0}}} \tag{18} \end{align}\] |
Equations (16) and (18) show that the parallel resonance frequency of a single-segment T-type LPM is greater than that of a \(\Gamma\)-type LPM. The ratio is \(\sqrt{\text{2}}\). This is consistent with the difference in analog accuracy.
3. Determination of Cascaded Number
3.1 Characteristic Impedance of LPM
Characteristic impedance is an inherent characteristic in a transmission line that affects the amplitude and phase changes of voltage and current, equal to the ratio of voltage to current at various locations. The characteristic impedance of the LPM also characterizes the analog accuracy. According to the transmission line theory, the parameters of the accurate equivalent line are expressed as
| \[\begin{align} \begin{cases} \displaystyle Z_{S} = Z_{c} \sinh(\gamma l) \\ \displaystyle Y_{P} = \frac{1}{Z_{c}} \tanh (\gamma l) \\ \end{cases} \tag{19} \end{align}\] |
The distributed parameter model in Fig. 1 can be equivalent through the \(\pi\)-type LPM with the same topology. The equivalent series impedance \(Z_{pi}\) and parallel capacitance \(Y_{pi}\) of the LPM are expressed as
| \[\begin{align} \begin{cases} \displaystyle Z_{pi} = \frac{j\omega L_{0}l}{m} \\ \displaystyle Y_{pi} = \frac{j\omega C_{0}l}{2m} \\ \end{cases} \tag{20} \end{align}\] |
Compared with (19) and (20), it can be concluded that
| \[\begin{align} \begin{cases} \displaystyle Z_{c\pi} = \frac{j\omega L_{0}l}{m} \bigg/ \sinh \left(j\omega\sqrt{L_{0}C_{0}}\frac{l}{m}\right) \\ \displaystyle Z_{c\pi} = \tanh \left(j\omega\sqrt{L_{0}C_{0}}\frac{l}{m}\right) \bigg/ \frac{j\omega C_{0}l}{2m} \\ \end{cases} \tag{21} \end{align}\] |
where \(Z_{c\pi}\) is the characteristic impedance of the \(\pi\)-type LPM. Solving (21) yields
| \[\begin{align} \sin({\omega l\sqrt{L_{0}C_{0}}}/{(2m)}) = {\omega l\sqrt{L_{0}C_{0}}}/{(2m)} \tag{22} \end{align}\] |
Therefore, \(Z_{c\pi}\) is
| \[\begin{align} Z_{c\pi} = \sqrt{\frac{L_{0}}{C_{0}}} \frac{1}{\displaystyle \sqrt{1 - \frac{\pi^{2}f^{2}l^{2}L_{0}C_{0}}{m^{2}}}} = Z_{c}k_{pi} \tag{23} \end{align}\] |
The \(k_{pi}\) is the characteristic impedance coefficient, which is the ratio of the characteristic impedance of the cascaded \(\pi\)-type LPM to that of DPM. The closer \(k_{pi}\) is to 1, the better the equivalence effect of the cascaded \(\pi\)-type LPM is. For the circuit with a fixed cascaded number, the low frequency has higher equivalent accuracy. The accuracy decreases as frequency increases. If high equivalent accuracy is required at the high-frequency band, it is necessary to increase the cascaded number. At 50 Hz, the \(k_{pi}\) of the single-cascade circuit is 1.005. This is very high accuracy. Therefore, to investigate the transmission characteristics of the line at the operating frequency, a single-cascade \(\pi\)-type LPM can be selected. It is simple and low-cost. For the frequency of 1000 Hz, the characteristic impedance coefficient is less than 1.01 when the cascaded number is larger than 18. Figure 8 (a) shows \(k_{pi}\) with different cascaded numbers and frequencies.
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Fig. 8 Characteristic impedance coefficient of LPMs. |
Similarly, the characteristic impedance of cascaded T-type LPM can be also obtained, and is expressed as
| \[\begin{align} Z_{cT} = \sqrt{\frac{L_{0}}{C_{0}}}\sqrt{1 - \frac{\pi^{2}f^{2}l^{2}L_{0}C_{0}}{m^{2}}} = Z_{c}k_{T} \tag{24} \end{align}\] |
The \(k_{T}\) is the characteristic impedance coefficient of the cascaded T-type LPM. According to (23) and (24), the characteristic impedance of DPM can be expressed as
| \[\begin{align} Z_{c} = \frac{Z_{c\pi}}{k_{pi}} = Z_{cT}k_{pi} \tag{25} \end{align}\] |
Equations (23) and (24) indicate that the characteristic impedance coefficient of cascaded T-type LPM is the reciprocal of that of cascaded \(\pi\)-type LPM. With the increase of cascaded number, their characteristic impedance coefficients are gradually close to 1. Equation (25) shows that although the characteristic impedances of LPMs are different, the ratios to characteristic impedance of DPM are the same. Therefore, the cascaded \(\pi\)-type LPM has the same accuracy with the cascaded T-type LPM.
As shown in Fig. 8, when the cascaded number is 1 and \(f > 500\), both \(k_{pi}\) and \(k_{T}\) are imaginary numbers. The LPMs cannot be used for simulation because the characteristic impedance of lossless lines is a positive real number. Therefore, the characteristic impedance of the cascaded LPM should also be a positive real number. For this reason, the cascaded number should satisfy
| \[\begin{align} m > \pi fl\sqrt{L_{0}C_{0}} \tag{26} \end{align}\] |
3.2 Cascaded Number
The modal information shows that the same first resonance frequency determines the same cascaded numbers of LPMs. Take the cascaded \(\pi\)-type circuit as an example. According to (23), when \(k_{pi}\) is larger than 1, a specific frequency \(f\) can be expressed as
| \[\begin{align} f = \frac{m\sqrt{k_{pi}^{2} - 1}}{\pi lk_{pi}\sqrt{L_{0}C_{0}}} \tag{27} \end{align}\] |
The ratio \(k_{0}\) of frequency \(f\) to the last resonance frequency is expressed as
| \[\begin{align} k_{0} = \frac{m\sqrt{k_{pi}^{2} - 1}}{\pi lk_{pi}\sqrt{L_{0}C_{0}}} \Bigg/ \frac{m}{\pi l\sqrt{L_{0}C_{0}}} = \frac{\sqrt{k_{pi}^{2} - 1}}{k_{pi}} \tag{28} \end{align}\] |
To improve analog accuracy, the maximum simulation frequency of the cascaded \(\pi\)-type LPM should be less than the first resonance frequency of a single segment. As can be seen from (14), the ratio of the first resonance frequency to the final resonance frequency is \(\sqrt{2}/2\). Therefore, the \(k_{0}\) of cascaded \(\pi\)-type LPM is less than \(\sqrt{2}/2\). So do cascaded T-type LPM. The first resonance frequency of \(\Gamma\)-type LPM is \(\sqrt{2}/2\) of the T-type equivalent circuit. The corresponding maximum \(k_{0}\) is 1/2. Therefore, the \(k_{pi}\) of the cascaded \(\Gamma\)-type LPM should be less than \(2\sqrt{3}/3\). The optimal cascaded numbers of cascaded \(\pi\)-type, T-type and \(\Gamma\)-type LPMs are shown in (29).
Where \(f_{max}\) is the maximum simulation frequency. The \(\lceil x\rceil\) is the upward rounding function of \(x\). When the simulation effect demand is not high, \(k_{pi}\) can be set to the
| \[\begin{align} m_{op} = \left\{ \begin{array}{@{}ll@{}} \displaystyle \left\lceil k_{pi} \pi f_{\max} l \sqrt{\frac{L_0 C_0} {k_{pi}^2-1}} \right\rceil & \mbox{$\pi$ (T)-type} \\ \displaystyle \left\lceil \sqrt{\frac{3}{2}} k_{pi} \pi f_{\max} l \sqrt{\frac{3L_0 C_0}{2(k_{pi}^2 - 1)}} \right. \\ \displaystyle \left. \quad= \frac{3}{2} k_{pi} \pi f_{\max} l \sqrt{\frac{L_0C_0}{k_{pi}^2 -1} } \right\rceil & \mbox{$\Gamma$-type} \end{array}\right. \kern-10pt \tag{29} \end{align}\] |
upper limit. When there is a high demand for simulation accuracy, the \(k_{pi}\) of the \(\pi\)-type LPM should be reduced. The accuracy of \(\Gamma\)-type LPM decreases in the same proportion.
For the transmission line in Table 1, the wavelength of 3000 Hz is 15.775 km. If the cascaded number is determined based on “1/30 of the wavelength”, a 57-segment LPMs is required. This is a fairly large number. Substituting \(m=57\) to (23) gives
| \[\begin{align} k_{pi}= \frac{1}{\displaystyle \sqrt{1-\frac{\pi^2}{30^2}}} = 1.0055 \tag{30} \end{align}\] |
Equation (30) indicates that the conventional model has a high simulation accuracy. However, for physical simulation systems of general distribution networks, such high simulation accuracy is not necessary. According to (23), when \(k_{pi} = \sqrt{2}\), the cascaded number of \(m=9\). Set the \(k_{pi}\) between the conventional cascaded number and \(\sqrt{2}\), such as \(k_{pi}=1.05\), and the corresponding cascaded number is \(m=20\). The variation curve of \(m\) with frequency for different stages is shown in Fig. 9.
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Fig. 9 Characteristic impedance coefficients with different cascaded numbers. |
It can be seen that when \(m=9\), the change of \(k_{pi}\) is significant within 3000 Hz. When \(m=20\) or \(m=57\), \(k_{pi}\) remains relatively close to 1. The cascaded number determined by the conventional method is excessive. For distribution network simulation systems with general accuracy requirements, the cascaded number can be determined based on the upper limit of \(k_{pi} = \sqrt{2}\).
4. Simulation Verification
According to (29), when \(k_{pi} = \sqrt{2}\), the cascaded number of \(\pi\)-type, T-type, and \(\Gamma\)-type LPMs are 8.4, 8.4, and 12.7, respectively. The rounded values are 9, 9 and 13, respectively. The line voltage is set to 5.77 kV. The topology of selected cascaded \(\Gamma\)-type LPM is shown in Fig. 6 (a). A 200 V step voltage disturbance is added at the head of the line at 1 s. A resistor of 10 \(\Omega\) of is connected at the end.
Perform impedance-frequency scanning at the end of the line, as shown in Fig. 10. Figure 10 shows that the resonance period of DPM is fixed, while the resonance periods of LPMs gradually decrease. This is consistent with the analysis results based on RMA. From (a) and (b), it can be seen that within 3000 Hz, the impedance characteristics of cascaded \(\pi\)-type, T-type, and \(\Gamma\)-type LPMs are close to that DPM. When the resonance frequency is greater than 3000 Hz, the difference in resonance characteristics gradually becomes apparent. For example, the \(f=3551\) Hz is the parallel resonance frequency of the DPM. However, it is close to the series resonance frequencies of \(\pi\) (T)-type LPM with 9 cascades and \(\Gamma\)-type LPMs with 13 cascades. Figure 10 (c) shows that the \(\pi\)-type and T-type LPMs have the same impedance characteristics. They have same parallel and series resonance frequencies. Even when \(f > 3000\) Hz, the phases of impedance of them are same. When \(f > 3000\) Hz, the impedances of \(\pi\)-type and T-type LPMs are remain inductive and capacitive, respectively. As shown in Fig. 10 (d), the \(\Gamma\)-type LPM with 9 cascades have lower accuracy. The \(f=2761\) Hz is the parallel resonance frequency of DPM. But it is close to the series resonance frequencies of \(\Gamma\)-type LPM with 9 cascades. This opposite phenomenon also occurs at \(f=3000\) Hz.
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Fig. 10 The results of impedance-frequency scanning. |
For the cascaded \(\pi\)-type model, Fig. 11 shows the simulation results with different \(m\). Figure 11 shows that as \(m\) increases, the cascaded model gradually becomes similar to the transmission line. Within 3000 Hz, the result calculated based on \(k_{pi} = \sqrt{2}\), i.e. \(m=9\), also has a good simulation effect. There is no significant difference between the driving point impedances of \(m=20\) and \(m=57\) and the driving point impedance of the transmission line. These four models exhibit four parallel resonances within 3 kHz. When the frequency exceeds 3000 Hz, the model with \(m=9\) deviates significantly from the transmission line. The model with \(m=20\) shows significant differences from the transmission line, and the model with \(m=57\) is still consistent with the transmission line. Therefore, the proposed method meets the simulation requirements of impedance characteristics while reducing costs.
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Fig. 11 Frequency characteristics of transmission lines. |
To understand the current response more clearly, the end of the line is connected with a resistance of 1000 \(\Omega\). The current wave at the headline is shown in Fig. 12. It can be seen that three lumped models have accurate simulation effects. Perform fast Fourier transform (FFT) for one cycle after 1 s. As can be seen from Fig. 13, within a range of less than 3000 Hz, the harmonic content of the distributed parameter model and LPMs are close. DPM and LPMs have significant voltage distortion at 8, 23, 37 and 49 pu. For 49 pu, the harmonic voltage of \(\pi\)-, T-, \(\Gamma\)-type LPMs and DPM are 12.7, 11.5, 11.2 and 11 V, respectively. In frequency band above 3000 Hz, the difference gradually becomes obvious, as shown in Fig. 13. DPM has obvious voltage distortion at 65, 76, 86, and 95 pu. However, \(\pi\)-, and T-type LPMs have obvious voltage distortion at 67 and 73 pu. For \(\Gamma\)-type LPM, the resonance frequencies are 61 and 70 pu.
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Fig. 12 The head current of distributed parameter model and LPMs. |
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Fig. 13 FFT results of the voltage at the end of line. |
5. Conclusions
This paper proposes a determination method of the cascaded number of the transmission line. The conclusions obtained through theoretical and simulation analysis are as follows.
- 1) The resonance characteristics of DPM and LPMs are discussed based on RMA and numerical analysis. The resonance frequency and input impedance of transmission lines are periodic. The resonance frequencies of the cascaded \(\pi\)-type and T-type LPMs are equal to the cascaded number, and each resonance frequency is smaller than that of DPM. The number of the resonance frequency of open-circuit \(\Gamma\)-type LPM is equal to the cascaded number minus 1. The number of resonance frequencies of short-circuit \(\Gamma\)-type LPM is equal to the cascaded number. The first resonance frequency characterizes the accuracy of the cascaded LPM.
- 2) The characteristic impedance coefficients of cascaded \(\pi\)-type LPM and T-type LPM are reciprocal to each other. The LPMs with the same cascaded number of T-type and \(\pi\)-type LPMs have the same accuracy. The accuracy of \(\Gamma\)-type LPM is smaller than that of \(\pi\)-type LPM and T-type LPM with the same cascaded number. The cascaded number for LPMs based on the first resonance frequency of single segment is obtained.
Acknowledgments
This work was supported by the National Key R&D Program: 2019YFE0118000.
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Authors
Yunnan Power Grid Co. Ltd.
graduated from Guangxi University, Nanning, China, majoring in Power System and Automation. He was employed by Electric Power Science Research Institute of Yunnan Power Grid Co. Ltd. His research interests include power quality management and model predictive control.
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Yunnan Power Grid Co. Ltd.
graduated from Kunming University of Science and Technology, Kunming, China, majoring in Power System and Automation. He was employed by Production Technology Department of Yunnan Power Grid Co. Ltd. His research interests include voltage and reactive power management, and control of power converters.
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Yunnan Power Grid Co. Ltd.
graduated from North China Electric Power University, Beijing, China, majoring in Power Systems and Automation. He was employed by Electric Power Science Research Institute of Yunnan Power Grid Co. Ltd. His research interests include power quality management and multilevel converters.
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Yunnan Power Grid Co. Ltd.
graduated from Taiyuan University of Technology, Taiyuan, China, majoring in Power Systems and Automation. He was employed by Electric Power Science Research Institute of Yunnan Power Grid Co. Ltd. His research interests include power system analysis and grid-tied power converters.
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Yunnan Power Grid Co. Ltd.
graduated from Honghe University, Honghe, China, majoring in Electrical Engineering and Its Automation. She was employed by Honghe Power Supply Bureau of Yunnan Power Grid Co. Ltd. Her research interests include the control strategy of reactive power and voltage in the distribution network.
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Yunnan Power Grid Co. Ltd.
graduated from Wuhan University, Wuhan, China, majoring in Power System and Its Automation. He was employed by Dali Power Supply Bureau of Yunnan Power Grid Co. Ltd. His research interests include power quality management and grid-tied power converters.
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