Application of bypass damping filter in suppressing subsynchronous resonance of multi-generator series-compensated systems

Electric Power Systems Research 168 (2019) 117–126

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Application of bypass damping filter in suppressing subsynchronous resonance of multi-generator series-compensated systems

T

Shijia Wang, Zheng Xu , Facai Xing ⁎

Department of Electrical Engineering, Zhejiang University, Hangzhou, Zhejiang Province, PR China

ARTICLE INFO

ABSTRACT

Keywords: Bypass damping filter (BDF) Subsynchronous resonance (SSR) Torsional interaction Transient torque amplification Multi-generator series-compensated system

Subsynchronous resonance (SSR) is one of the main risk factors for ac power transmission systems with series compensation. In bulk power transmission from generator groups, the diversity of shaft torsional modes of different generators makes the SSR suppression difficult, especially for the scenario where different types of generators are concentratedly connected. In this paper, it is shown that the bypass damping filter (BDF) can be applied to suppress SSR for series-compensated system with multi-type generators. An objective optimization method for the BDF parameter tuning is proposed. This method, with clear physical meaning and application convenience, considers the torsional stability criterion of all objective generators. The SSR damping effectiveness of the BDF with the proposed method is verified in a twenty-generator series-compensated system, through timedomain simulation and torsional mode analysis. The result shows that the BDF can be an effective measure for suppressing both the torsional interaction effect and the transient torque amplification effect.

1. Introduction Applying series-compensation in long-distance ac transmission line is an effective method to increase power transmission capacity between sending and receiving power grids. However, series-compensated lines may cause subsynchronous resonance (SSR), leading to coupled oscillation between mechanical system and electrical system at frequencies lower than the system rated frequency. In more severe cases, turbinegenerator shafts may be damaged by torsional oscillation caused by unreasonable capacitor configuration [1,2]. Ever since SSR occurred for the first time in America in the 1970s [3], many suppression methods have been proposed and they can be categorized as shown in Fig. 1. One category is to use supplementary SSR controllers on generator excitation system, for instance the supplementary excitation damping controller (SEDC) [4–7] and the modification of PSS [8,9]. These supplementary controllers usually adopt shaft speed deviation (Δω) as input signal and output excitation voltage increment through amplification and phase compensation. The other category is to install auxiliary devices, and it can be further divided into two subcategories according to electrical connection type between the device and the investigated generator, namely the parallel-connected devices and the series-connected devices. Typical parallel-connected devices include the static var compensator (SVC) [10,11], the static synchronous compensator (STATCOM) [12–14], etc. Similar to the

⁎

supplementary controllers, the parallel-connected devices also use Δω as input, but they output reactive power increment so that the connected bus voltage can be properly adjusted to damp SSR. The seriesconnected devices are usually installed on series-compensation line and they do not necessarily require Δω as input (hence dotted line is used in Fig. 1). These devices are able to change line impedance in the subsynchronous frequency range temporarily (e.g., the thyristor-controlled series capacitor [15–17]) or permanently (e.g., the block filter [18]), so that subsynchronous currents at certain frequency can be directly damped. Among the existing SSR problems, SSR in multi-generator seriescompensated system is relatively more complicated. In this particular scenario, different types of generator shafts will lead to the dense distribution of shaft torsional modes within the subsynchronous frequency range. The aforementioned suppression methods can partly handle the multi-generator SSR problem, but shortcomings still exist. As for the supplementary SSR controllers, the quantity of controllers needed is large, for one controller usually treats merely one generator. For example, in Ref. [6], the SEDC is applied on each of the generator in a practical four-machine system. Besides, the supplementary control is generally effective in suppressing torsional interaction effect (TIE), which is one phenomenon of SSR; but its suppression effect on another phenomenon of SSR, the transient torque amplification effect (TTAE), is not satisfactory, for it cannot directly damp the subsynchronous

Corresponding author. E-mail address: [email protected] (Z. Xu).

https://doi.org/10.1016/j.epsr.2018.11.010 Received 29 May 2018; Received in revised form 9 September 2018; Accepted 13 November 2018 0378-7796/ © 2018 Published by Elsevier B.V.

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Nomenclature

XT Ra Rr Xd ″ Δθ, Δω Ke, De ψ0 H δm, δe, δ

All of the electrical variables in this paper are in per-unit values fe, fm XC RLine XLine RBDF XBDF

Electrical frequency and its counterpart mechanical frequency, fm = 1 − fe Capacitive reactance of series capacitor at the rated frequency Resistance of series compensation line Inductive reactance of series compensation line at the rated frequency Resistance of BDF Capacitive/inductive reactance of BDF at the rated

i j

frequency Transformer leakage reactance at the rated frequency Generator armature resistance Generator rotor resistance Generator subtransient reactance Increment of rotor angular displacement and speed Coefficient of synchronizing torque and electrical damping Generator flux linkage under steady state Torsional mode inertial constant Torsional mode mechanical, electrical and total damping factor Subscript represents the i-th generator Subscript represents the j-th torsional mode

Fig. 1. Three categories of SSR suppression methods.

currents. Similar shortcomings also exist for the parallel-connected devices, for instance in the case study of Ref. [19], four SVCs are configured in a six-machine system to suppress SSR, and the initial amplitudes of transient torques are not decreased after SVCs in service. The series-connected SSR suppression devices are installed on series-compensated lines, therefore are able to damp the subsynchronous currents directly and should be effective in suppressing both TIE and TTAE for multi‐generator system. However, for this kind of devices, in engineering practice the multi-modal of different types of generator shafts will lead to the difficulty in parameter tuning [18]. The bypass damping filter (BDF) is also in series-connection with the target generator. Initially, the BDF is considered effective in restraining only one phenomenon of SSR, the induction generator effect (IGE) [20,21]. Refs. [22,23] further indicate that the BDF is able to suppress both TIE and TTAE; and a BDF parameter tuning method is proposed in Ref. [22] based on a single-generator system, which quantitatively explains the SSR suppressing mechanism of the BDF. However, due to the complexity of grid structure, the method in Ref. [22] cannot be directly applied for multi-generator systems. This paper aims at extending the application of BDF in multi-generator series-compensated systems, by proposing a new BDF parameter tuning method. This method considers the shaft torsional stability of each torsional mode in each generator and is executed through nonlinear optimization. With this method the effectiveness of the BDF is verified in a twenty-generator system with different shaft structures. Contribution of this study can be summarized as follows.

of BDF on suppressing SSR as preliminary knowledge, which helps explain the reason why BDF theoretically contributes to mitigating IGE, TIE and TTAE. In Section 3, the BDF parameter tuning method for multi-generator series-compensated systems is proposed, combined with the torsional stability analysis for torsional modes of all investigated generators. In Section 4, the proposed method is verified in a twenty-generator series-compensated system derived from a power base transmission system of northern China, using time-domain simulation and torsional mode analysis. The main conclusions of this paper are presented in Section 5. 2. Mechanism of BDF on suppressing SSR As shown in Fig. 2, a BDF, which is parallelly connected with a series-capacitor, consists of a damping resistor and a parallel combination of a reactor and a capacitor. The mechanism of BDF on suppressing the three phenomena of SSR, namely IGE, TTAE and TIE, will be explained below. 2.1. Mechanism of suppressing IGE and TTAE The parallel combination of the reactor and the capacitor in the BDF is designed to be resonant at the system rated frequency so that its impedance at the rated frequency is high. In normal conditions, the BDF carries virtually no current; hence it does not influence the steady-state

• The effectiveness of BDF on suppressing SSR is verified and extended beyond simple power system. • A parameter tuning method of BDF is proposed, which is of clear •

physical meaning and can be conveniently applied. This method is able to solve the three aspects of SSR (IGE, TIE and TTAE) in multigenerator series-compensated system. A complicated test case based on real system is provided without reservation, which can be used for follow-up studies.

This paper proceeds as follows. Section 2 introduces the mechanism

Fig. 2. Structure and connection mode of BDF. 118

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operation of the system. At subsynchronous frequencies, due to the decrease of the impedance of the parallel combination, the BDF becomes a resistive/inductive bypass path for the subsynchronous currents. The resistive/inductive bypass path can partly or wholly offset the generator negative resistance (see Ref. [1] for further explanation) at subsynchronous frequencies; therefore it contributes to the mitigation of IGE. As for TTAE, its severity depends on the disturbance energy injected into the shaft system during fault period. Considering that the resistive/inductive bypass path helps damp the subsynchronous currents, or in other words consume the disturbance energy from subsynchronous currents, the BDF also has potential for suppressing TTAE.

Except for special illustration, all the electrical parameters below are expressed in per-unit value. 3.1. Step 1: Generator equivalence A power plant in the investigated system usually has generators of the same type. These generators have identical shaft structure and symmetric electrical connection, and are usually in identical operation condition. In this step, generators of the same type from the same power plant are equated as one generator, using the equivalent method in Ref. [24]. After Step 1, suppose the equivalent system has N nodes and n generators. For simplification, the number of these generator nodes is successively marked as 1, 2,…, n. Further suppose that the shaft of the ith generator has mi torsional modes, with typical mechanical frequencies denoted as fm (ij) (i = 1, …, n, j = 1,…, mi). The counterpart electrical frequency are denoted as fe (ij) (=1 fm (ij) ) .

2.2. Mechanism of suppressing TIE The occurrence of TIE largely depends on electrical damping. If severely negative electrical damping appears at frequencies close to typical frequencies of the shaft system, and cannot be offset by mechanical damping, the TIE is likely to occur. The suppression mechanism of the BDF on TIE can be explained through electrical damping analysis. Take the series-compensated system in Fig. 2 as an example. Suppose the per-unit system impedance at the rated frequency is Z = R + jXL − jXC, where R is the system net resistance, XL is the total inductive reactance and XC is the capacitive reactance. Without BDF the electrical damping will reach the negative extremum at series-resonance frequency fe = XC / XL . The counterpart mechanical frequency fm (=1 − fe, expressed in per-unit value) may approach one of the typical frequencies of the shaft system, making the negative electrical damping cannot be offset by the mechanical damping. As a result, shaft torsional instability (namely TIE) will take place. The application of the BDF changes the system impedance within subsynchronous frequency range, hence the frequency where negative electrical damping reaches minimum will be altered, deduced in Ref. [22] for a single-generator system (see Fig. 2) as

fm = 1

fe = 1

XC (XBDF + XL ) XL (XBDF + XC )

3.2. Step 2: Node impedance scanning This step is based on the inversion of the system node admittance matrix (SNAM), and should be executed successively for each torsional mode of each generator. In order to acquire the impedance seen from node i without the influence of the i-th generator, a node voltage equation is defined as

[Y (fe )

(2)

Yi (fe )] U = I

where U and I represent column vectors of node voltage and node injection current; Y(fe ) denotes the complete SNAM generated at frequency fe; Yi (fe ) represents influence of the i-th generator. For Yi (fe ) , only its i-th diagonal element equals the reciprocal of the i-th generator impedance at fe, while other elements equals zero. The influence of the BDF on the system is reflected in the formation of Y(fe ) , where the BDF is considered in calculating the impedance of the connected SCL.

ZSCL (fe ) = RLine + jfe XLine +

(1)

jXC //ZBDF (fe ) fe

(3)

The symbol “//” in (3) represents impedance calculation of parallelconnected components, and the BDF impedance at fe is denoted as

It means that XBDF can be purposively adjusted, so that the negative extremum of electrical damping appears where the typical frequencies of the shaft system are avoided, hence TIE can be suppressed.

ZBDF (fe ) = RBDF +

3. BDF parameter tuning for multi-generator series-compensated systems

As a result, the impedance seen from node i at frequency fe (ij) , denoted as Zij, can be acquired as

Zij = IiT [Y (fe (ij) )

In Ref. [22], a BDF parameter tuning method has been proposed for a single-generator series-compensated system, based on the extremum seeking of electrical damping. But this method is no longer feasible for a multi-generator series-compensated system. This is because, in such a complex system, a generator usually has several rather than only one extremum of electrical damping, the assumption of this method is not valid. Besides, the proposed method in Ref. [22] considers the torsional stability of only one target generator, but in a multi-generator system the torsional stability of all the generators should be considered together. In this section, a BDF parameter tuning method, which comprehensively considers the torsional stability of each investigated generator, will be proposed. The proposed method consists of the following steps and will be illustrated with the theory of shaft torsional criterion. Note that, if the aimed system has more than one series-compensated line (SCL), the installation location of the BDF should be firstly decided. The location selection of the BDF will be discussed in Section 4 combined with a detailed system. In this section, it will be assumed that the BDF location has been settled.

jfe XBDF 1

fe2

Yi (fe (ij) )] 1 Ii

(4)

(5)

where Ii is a N × 1 column vector, its i-th element equals one, while others equal zero. The influence of FACTS devices can also be reflected in the formation of Y(fe ) . 3.3. Step 3: Electrical damping calculation The dynamics of generator shaft can be regarded as a combined response of mechanical torque Tm and electrical torque Te; As turbine dynamics is inherently slow [25], Tm can be considered constant for SSR analysis. The deviation of Te can be expressed as

Te = K e

(6)

+ De

De greatly influences shaft stability. In this step, De related to the j-th torsional mode of the i-th generator is calculated as [21]

De (ij ) =

119

2 0(i )

fe (ij) Real(Zij + ZG (ij) ) 2fm (ij)

Zij + ZG (ij ) 2

(7)

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where Real(·) means the real part; ZG (ij) is the impedance of the i-th generator at fe (ij) , expressed as [26]

ZG (ij) = Ra (i) +

Rr (i) s

+ jfe (ij ) Xd (i)

minimum damping factor, the application of the BDF should make the minimum damping factor as large as possible. Hence the objective function for the BDF parameters tuning is chosen as

(8)

max minp

where s is the slip factor and Rr (i) is the effective rotor resistance, they are denoted as

s=

fe (ij)

1

fe (ij)

Rr (i) =

1 2

X d (i ) B Td 0

fe (ij) +

(9)

X q (i ) B Tq0

(10)

ωB is the base value of angular speed. 3.4. Step 4: Damping factor calculation

ij

=

4Hij

+

m (ij)

=

e (ij)

+

m (ij)

i = 1, … , n [ ij (RBDF , j = 1, … , mi

XBDF , p)] (12)

• The BDF parameters acquired should be practically reasonable. That

In this step, the damping factor for the j-th torsional mode of the i-th generator is calculated as [21]

De (ij)

min

where P is the set of potential system states, for instance, N-1 state of parallel-connected transmission lines (see Subsection 4.1 for better understanding); but if the system has only one state to be considered, the minp∈P part can be ignored. The (RBDF , XBDF , p) is specially written to emphasize that δij is determined by the BDF parameters and the given system state. The objective function (12) is actually a nonlinear programming, and can be solved by genetic algorithm (GA) [27]. Some key points are listed as follows, while other execution details will be discussed combined with case study in the next section.

fm (ij)

=

P

(11)

•

In engineering practice, a torsional mode is considered stable if its damping factor is positive. The m (ij) depends on the load level of the generator and is usually about 0.03–0.10; the e (ij ) , as can be seen in Step 3, largely depends on the system impedance and will be comprehensively considered in the next step. 3.5. Step-5: BDF parameters tuning

•

It can be found in the above steps that, for each of the investigated generators, the parameters of the BDF directly influence its node impedance, and will further affect the generator electrical damping as well as the final damping factors. In other words, the damping factors of the shaft torsional modes can be seen as the function of the BDF parameters if the system condition p is determined (except for the BDF parameters). As the torsional stability of a generator group usually depends on the

is to say, these parameters can be implemented physically. As a result, before the execution of GA, the numerical range of RBDF and XBDF should be determined. The GA has some probability (actually very small) to find partial optimal solution. This possibility can be avoided by resetting the initial population and re-executing the GA to check the consistency of results (or find the best result). But a partial solution may also be useful. From Eq. (11), a complex multi-generator system is torsionally stable if the damping factors of all the torsional modes are positive. An acquired BDF parameter, no matter partial or global optimal, is feasible if the above target can be satisfied. Some torsional modes may still have negative damping factors after the optimization process. Two possible reasons and relevant solutions are as follows: 1) The preset BDF parameter range is too narrow, especially when the acquired BDF parameter touches the lower or upper parameter boundary. In this condition, the preset BDF parameter range should be correspondingly expanded and the optimization

Fig. 3. Test system with BDF applied. 120

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Table 1 Results of BDF parameter tuning. XC (Ω)

Lower limit (Ω)

Upper limit (Ω)

Result (Ω)

RBDF-1 XBDF-1

28.0 (C1)

2.8 8.4

14.0 28

9.8 10.6

RBDF-2 XBDF-2

37.2 (C2)

3.7 11.2

18.6 37.2

12.1 15.7

•

be re-executed. 2) The stability of certain torsional modes is difficult to be improved by BDF. In this condition, these torsional modes should be exempted from the optimization, and extra means are needed to keep them stable. For instance, installing parallel-connected devices or applying supplementary controls for the relevant generators, as has been mentioned in Section 1. The computational complexity of GA in this scenario is controllable. The complexity of GA is O(g · s · m · n3), where g is number of generations, s is the population size, m is the chromosome bits and n is number of system buses. The n3 represents the fitness calculation for individuals, which needs matrix inversion with complexity O(n3) (see Eq. (5)). The g, s and m can be manually adjusted according to precision requirement of BDF parameters. The number of system buses (n) is decided by system scales; in Ref. [28], a detailed method has proved that system scales can be largely reduced with guaranteed precision in SSR analysis.

4. Case study In this section, the effectiveness of the BDF with the proposed parameter tuning method will be verified in a twenty-generator seriescompensated system, based on torsional mode analysis and time-domain simulation. As shown in Fig. 3, the test system, derived from a thermal power base transmission system in the northern grid of China, is a typical power transmission grid. Some system parameters have been marked on Fig. 2 and others are listed in Appendix. The sending end of the system includes three power plants, with 8, 8 and 4 generators, respectively. These generators have three rated values of active power and eight types of shaft structure; each shaft structure has three torsional modes. The power is transmitted through EHV and UHV lines, among which three UHV lines are series-compensated. The receiving end is represented by three equivalent systems (ESs), each ES receives active power of about 5000 MW.

Fig. 4. Flow path of GA.

mainly for Plant-A & C and also acts as a backup for Plant-B. Note that, for better suppression effect, a BDF should be applied to each line if several SCLs are in parallel-connection. 4.1.2. Solving BDF parameter Solving BDF parameters relies on the proposed method in Section 3 combined with GA. Before executing the algorithm, the numerical range of BDF parameters (RBDF and XBDF) and the potential system conditions should be decided. As for the numerical range of BDF parameters, it is shown in Eq. (4) that the damping component of BDF (RBDF) will be more effective if XBDF is relatively small. However, smaller value of XBDF means larger capacitance and higher cost. In Ref. [18], the chosen RBDF and XBDF is about 1/5 and 1/2 of the capacitive reactance of the series-capacitor, respectively. Hence for this test system, the numerical range of RBDF and XBDF is respectively chosen as [0.1XC, 0.5XC] and [0.3XC, 1.0XC], where XC represents capacitive reactance of the series-capacitor at the rated frequency. The details of the numerical range are shown in Table 1, note that the preset numerical range is chosen a little wide so that the optimal result can be covered. As for the potential system states (see the symbol “P” in Eq. (12)), except for normal state, N-1 contingency on the three double-circuit or triple-circuit SCLs (line 3–5, line 5–6 and line 6–7) also are considered. Now that the location, the parameter range of the BDFs and the potential system conditions have been determined, the GA can be executed to acquire proper BDF parameters that satisfy the objective function (12). In the execution process, generators with identical shaft structure have been equated (for instance, A-1/2, A-4/5, A-6/7 and B1/2, see shaft parameters in Appendix A). ψ0 is estimated as 1.1 pu. The mechanical damping factor for all generators is set as 0 (zero), so as to highlight the electrical damping factor. This algorithm is coded on MATLAB and relevant details are shown as follows.

4.1. BDF configuration The BDF configuration consists of three parts, namely deciding BDF location (namely which SCL should install a BDF), solving BDF parameter and preliminary verification. 4.1.1. Deciding BDF location According to the mechanism analysis in Section 2, the function of the BDF is directly reflected in damping subsynchronous currents that injected into the generators during transient period. Therefore, deciding the location of the BDF should be based on the given grid structure. The main principle is that, for most of bus short‐circuit faults, the installed BDF should exist in the flow paths of the fault currents that injected into the generators from relevant fault points. Another empirical principle is that, the suppression effect is better if the BDF is closer to the generators. Hence the priority of installing a BDF should be given to seriescapacitor closer to the generators. Based on the principles above, two BDFs are applied for two SCLs (line 3–5 and line 5–6), denoted as BDF-1 and BDF-2, respectively. The BDF-1 aims at handling the SSR problem for Plant-B; the BDF-2 is 121

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Fig. 5. Comparison of calculated electrical damping factors. The blue, orange and yellow dots represent the first, second and third torsional mode of generators, respectively. (a) Normal operation. (b) N-1 contingency on line 3–5. (c) N-1 contingency on line 5–6. (d) N-1 contingency on line 6–7.

1) Precision. Considering the engineering practice, the precision of the BDF parameter is 0.1 Ω. Binary encoding length is chosen 8 bits. Note that the 8 bits satisfy BDF parameter range of 28 × 0.1 ≈ 26 Ω, which is sufficient for the preset BDF boundary. 2) Population size. The population size is chosen as 10, each individual has four 8-bit chromosomes, corresponding to RBDF1, XBDF1, RBDF2 and XBDF2, respectively. 3) Tolerance. The tolerance is chosen as 0.0001, which means that the GA will be terminated if the difference of minimum damping factors from two successive iterations is < 0.0001. 4) Fitness function. Each torsional mode has specific electrical damping factor e(ij) under specific BDF parameter. Considering that the minimum electrical damping factor without BDF is slightly larger than −0.1 (see Fig. 4(a)), the fitness function is chosen as

f (BDF) = 0.1 + minp

P

min i = 1, … , n [

e(ij)

1) Replication, Crossover and Mutation. The Roulette Wheel Selection is adopted to decide the replication probability of parents. One-point crossover operator is adopted as the chromosome length (8 bits) is short. The bit mutation probability is set to 1%. The replication, crossover and mutation are strictly consistent with those in Ref. [27]. 2) Flow path. The flow path of GA is shown in Fig. 4. The acquired result is given in the 5th column of Table 1. Note that the results in Table 1 do not touch the lower and upper limits, meaning that the preliminarily chosen range is wide enough to find proper BDF parameters. 4.1.3. Preliminary verification For comparison, the electrical damping factors for each torsional mode of each generator are plotted in Fig. 5. According to Eq. (11), a torsional mode is stable if e (ij ) > m (ij) . Therefore, the reverse of mechanical damping factor range [−0.10 to 0.03], marked as black solid lines, can be seen as “risk region”. The electrical damping factor that lies in or are very close to the risk region are also marked, for instance, B-6(2) corresponds to the 2nd torsional mode of generator B-

(BDF, p)]

j = 1, … , mi

(13)

The “0.1” is added to keep the fitness function positive.

122

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Fig. 6. Response comparison of shaft LP-GEN segment torques after a three-phase short-circuit fault. Y-axes represent per-unit torque values. Shaft torques of generators from the same plant are plotted in same color.

6. Fig. 5 shows that, without BDF applied, torsional modes can be found in or close to the risk region in each system condition; and the found torsional modes are different with the variation of system conditions. After the BDF applied, the improved electrical damping factors are all beyond the risk region. Actually, the minimum electrical damping factors in the four conditions are all slightly larger than zero. The system becomes sufficiently stable and from the perspective of simplicity, no other supplementary controls or devices are needed. However, it should be noted that the electrical damping factor with the BDF applied usually have upper limit. For instance, in Fig. 5 the electrical damping factors of the 20–30 Hz torsional modes are less than 0.05. If much larger damping factors (namely larger margin) are required, supplementary controls or other devices can be considered for further

improvement. As the damping factor is a crucial reflection of TIE, the effect of the BDF on suppressing TIE can, to some extent, be proved in Fig. 5. However, it should be admitted that in the proposed method, the calculation of electrical damping factor δe (see Eq. (7) and (11)) is relatively crude compared to the complexity of real generator model. For instance, the proposed method cannot reflect the mutual influence among nearby generators with similar but not identical shaft structures. Besides, according to the analysis in Ref. [26], the calculated De is slightly severer (more conservative) compared to the result from the test signal method and the complex torque coefficient method. As a result, further verification is needed; and in the following subsections, the effect of the BDF on suppressing TIE and TTAE will be verified 123

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Table 2 Mode identification of shaft LP-GEN segment. No.

Mode (Hz)

Without BDF

With BDF −1

A (pu)

σ (s

13.02 22.77 28.16

0.201 0.174 0.163

12.73 21.61 26.45

A-4/5

TIE effect

A (pu)

σ (s

0.126 0.090 0.030

0.155 0.073 0.145

0.132 0.090 0.087

Y Y Y

0.272 0.144 0.345

0.174 0.097 0.048

0.203 0.061 0.090

0.172 0.087 0.082

Y Y Y

Y

14.98 24.92 28.89

0.284 0.703 0.014

0.130 0.070 0.088

0.225 0.216 0.018

0.145 0.096 0.081

Y Y

A-6/7

14.31 22.51 26.66

0.307 0.385 0.066

0.143 0.097 0.069

0.241 0.157 0.021

0.156 0.103 0.080

A-8

15.62 25.94 29.90

0.275 1.179 –

0.145 0.027 –

0.208 0.219 0.013

B-1/2

10.60 21.25 25.98

0.080 0.115 0.062

0.113 0.081 0.076

B-3

13.65 23.77 26.51

0.365 1.354 0.938

B-4

13.90 21.51 27.09

0.275 0.369 2.018

A-1/2

A-3

)

TTAE effect −1

No.

Mode (Hz)

)

Without BDF

With BDF −1

A (pu)

σ (s

)

−1

A (pu)

σ (s

TTAE effect

TIE effect

)

B-5

14.31 22.51 26.66

0.425 0.861 0.308

0.192 0.102 0.069

0.150 0.105 0.018

0.207 0.109 0.080

Y Y Y

Y Y Y

B-6

15.62 25.94 29.90

0.418 3.566 0.181

0.154 0.023 0.080

0.148 0.182 0.015

0.151 0.098 0.081

Y Y Y

Y

Y Y

B-7

13.02 22.77 28.16

0.273 0.448 1.337

0.156 0.087 0.082

0.106 0.051 0.144

0.159 0.090 0.086

Y Y Y

Y Y Y

Y Y Y

B-8

14.98 24.92 28.89

0.424 2.197 0.159

0.156 0.076 0.080

0.152 0.166 0.019

0.164 0.098 0.080

Y Y Y

Y Y

0.150 0.102 0.081

Y Y

Y

C-1

10.60 21.25 25.98

0.076 0.071 0.027

0.104 0.081 0.073

0.020 0.013 0.003

0.140 0.082 0.079

Y Y Y

Y

0.035 0.009 0.002

0.102 0.080 0.074

Y Y Y

C-2

13.65 23.77 26.51

0.322 0.613 0.312

0.136 0.056 0.058

0.240 0.227 0.078

0.142 0.096 0.088

Y Y Y

Y Y

0.139 0.085 0.059

0.135 0.133 0.050

0.124 0.091 0.089

Y Y Y

Y Y

C-3

13.90 21.51 27.09

0.249 0.206 1.418

0.180 0.089 -0.015

0.195 0.075 0.221

0.213 0.103 0.101

Y Y Y

Y Y Y

0.182 0.108 0.010

0.097 0.059 0.144

0.148 0.140 0.085

Y Y Y

Y Y

C-4

13.02 22.77 28.16

0.257 0.241 0.240

0.163 0.087 0.041

0.185 0.089 0.212

0.164 0.090 0.091

Y Y Y

Y

Y

Y

Note: “–” means undetected. Rapidly converged components are not listed, for they do not influence torsional stability.

torsional frequency and φ0 is the initial phase. A and σ, which reflect the severity of TTAE and TIE, are the most concerned in SSR analysis. Note that in Table 2, the mark “Y” in the “TIE Effect” column means that the damping factor is increased by over 5%. It can be found that, the damping factors for the majority of torsional modes are improved with the BDFs applied. Especially for generator A-8, B-4, B-6 and C-3, in the original system they all have a torsional mode that is slowly damped or even divergent; with the BDFs applied these torsional modes turn sufficiently stable. As for other several torsional modes with decreased damping factors after BDFs applied, their damping factors are still larger than 0.07, hence the stability can still be guaranteed. 4.3. Effect of BDF on suppressing TTAE

Fig. 7. Peak torque value comparison of shaft LP-GEN segments.

During the fault period, the shaft fatigue caused by torsional oscillation mainly relates to two factors, the peak value of the transient torque and its damping rate. The damping rate concerns the TIE and has been discussed in the previous subsection. The peak value of the transient torque is a reflection of TTAE. According to the qualitative analysis in Section 2, the damping resistor in the BDF helps dissipate energy from subsynchronous currents, hence should have the ability to suppress TTAE. This point of view has been preliminarily verified in Fig. 6. The result of the time-domain simulation shows that, with the BDFs applied, the amplitudes of the transient torques for the majority of generators decrease significantly. This observation can also be proved in Table 2, where “Y” is marked in the “TTAE Effect” column if the peak value of transient torque drops by over 5%. It reveals that, for the vast majority of torsional modes, the application of the BDF contributes to the decline of their initial amplitudes. A more direct comparison of peak torque values for shaft LP-GEN segments is shown in Fig. 7. As SSR will not be stimulated if transmission line is not compensated, the maximum torque values from the condition of no compensation is also plotted (line reactance modified to

through time-domain simulation in PSACD/EMTDC combined with torsional mode analysis. 4.2. Effect of BDF on suppressing TIE The normal operation (no SCL is out of service) is simulated in PSCAD/EMTDC. The parameters in the simulation model are strictly consistent with those in Fig. 3 and Appendix A. All the generators operate at rated condition with δm = 0.08. Suppose a three-phase short-circuit fault occurs on BUS-6 at 1.0 s, the fault lasts 0.1 s. The comparison of shaft LP-GEN segment torques for all generators is shown in Fig. 6. It shows that the application of the BDF contributes to the convergence of shaft torques. Especially for generator C-3, its transient torque turn stable after BDF applied. In order to further demonstrate the effectiveness of the BDF on suppressing TIE, the torsional modes of the shaft torques in Fig. 5 are identified using the Prony method [29], the results are listed in Table 2. Generally, a torsional mode can be expressed as Ae t cos(2 fm t + 0) , where A is the initial amplitude, σ is the damping factor, fm is the 124

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maintain steady-state power flow). The black line represents steady state torque value (0.9 pu). Fig. 7 demonstrates that, for the LP-GEN segment of all the tested generators, their peak values of torques are suppressed after the BDFs applied. The suppressed values are even, for the majority of generators, slightly less than those with no series compensation. It should also be noted that the generators of Plant-B, which are electrically closer to the BDF, acquire superior suppression effect compared to the other generators. This phenomenon is in accordance with the BDF principle mentioned in Subsection 4.1.

The proposed method, which comprehensively considers the torsional stability of all investigated generators, has clear physical meaning and suits for engineering implementation. The effectiveness of the BDF on suppressing SSR is verified through time-domain simulation and torsional mode analysis. The result shows that, for a complex seriescompensated system, the BDF can help improve the torsional oscillation damping factors as well as reduce the peak torque values for the vast majority of investigated generators. Acknowledgement

5. Conclusion

This paper is supported by Headquarter Science and Technology Project of State Grid Corporation (Estimating, Suppressing and Preventing Control Technologies for Sub- and Super-synchronous Oscillations from New-energy Base under Complex Network Conditions).

This paper extends the application of the bypass damping filter in suppressing SSR of series-compensated bulk power transmission systems with different types of generator groups. The principle of the BDF location selection and the relevant BDF parameter tuning are proposed. Appendix A

The Appendix is a supplement to the parameters of the test system in Fig. 3. Shaft parameters Table A1 lists the shaft parameters. Note that Type 600-X, 660-X, and 1000-X represent rated active power 600 MW, 660 MW, and 1000 MW, respectively; the “X” represents subtype. Table A1 Shaft parameters. No.

Type

Typical frequency (Hz)/mode inertial (s)

A-1/2 A-3 A-4/5 A-6/7 A-8 B-1/2 B-3 B-4 B-5 B-6 B-7 B-8 C-1 C-2 C-3 C-4

600-1 600-2 600-3 660-1 660-2 1000-1 1000-2 1000-3 660-1 660-2 600-1 600-3 1000-1 1000-2 1000-3 600-1

13.02/2.10 12.73/1.78 14.98/1.70 14.31/1.42 15.62/1.53 10.60/8.62 13.65/2.26 13.90/1.26 Same as A-6/7 Same as A-8 Same as A-1/2 Same as A-4/5 Same as B-1/2 Same as B-3 Same B-4 Same as A-1/2

22.77/4.54 21.61/3.19 24.92/2.37 22.51/2.19 25.94/2.04 21.25/49.8 23.77/2.80 21.51/1.89

28.16/2.73 26.45/5.34 28.89/21.4 26.66/21.5 29.90/16.7 25.98/549 26.51/10.2 27.09/1.60

Generator electrical parameters All Generators use the following per-unit parameters in Table A2. In rated conditions, all generators have power factor of 0.9. Rated generator terminal voltage (L-L) is 20 kV for 600-MW and 660-MW generator, and 27 kV for 1000-MW generator. Table A2 Generator electrical parameters. Ra = 0.002518 Xd = 0.301

XP = 0.09595 Xq = 0.448

Xd = 2.155

Xd = 0.223

Xq = 2.1

Xq = 0.218

Td0 = 8.61 s

Tq0 = 0.956 s

Td0 = 0.045 s

Tq0 = 0.069 s

Transformer parameters The leakage reactance for all transformers is set to 0.16 pu. The capacity of generator step-up transformer is the same with the generator connected to it, namely 667 MVA, 732 MVA or 1120 MVA. Other three connection transformers have capacity of 5600 MVA.

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Line parameters Table A3 lists the transmission line parameters. In the execution of the proposed method, transmission lines are modeled using the lumped parameter π model, which is of sufficient accuracy in the subsynchronous frequency range. Table A3 Transmission line parameters. Type

Resistance (Ω/km)

Reactance (mH/km)

Capacitance (μF/km)

1000 kV

0.00685 0.0557

0.8403 2.674

0.01492 0.01015

500 kV

0.0202 0.1602

0.8753 2.715

0.01289 0.00877

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