{'fraction_non_alphanumeric': 0.034288435590126204, 'fraction_numerical': 0.013572505754424955, 'mean_word_length': 4.602489995553579, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Fast Method to Determine Short-circuit Capacity Based on Compensation Impact Factor

This paper proposes a fast method to determine short-circuit capacity for an AC system, which includes a compensation capacitor, to avoid sub-synchronous resonance (SSR) risk at a system planning stage. Firstly, the compensation impact factor (CIF) is calculated for generators with different short-circuit capacities. If CIF of all generators are below the threshold value, then the short-circuit capacity is the safety one. Finally, safety short-circuit capacity group is formed and strongly suggested to the system planning due to no SSR risk threating the system. The proposed method needs simple system modeling and calculation process and little information on generator shaft. Simulation results show that the proposed method can precisely determine the safety short-circuit capacity group for the system. Keywords—sub-synchronous resonance; short-circuit capacity; compensation impact factor


INTRODUCTION
Series compensation is a widely used technology in power system due to its advantages of enhancing power transmission capacity and transient stability in an economic way [1][2][3]. However, generator shaft of a thermal plant has a great opportunity to exchange sub-synchronous energy with the series compensation capacitor and then sub-synchronous resonance (SSR) occurs [4][5][6][7][8]. Since electric strength between AC Buses, which is decided by short-circuit capacity of an equivalent system, can change the resonance frequency of the system and damping characteristics of SSR [9][10][11][12], at the system planning stage it is critical to ensure the short-circuit capacity does not make contributions to exciting SSR.
The published SSR analysis method [13][14][15][16][17][18] needs complex computation process and detail data information of generators which is not available at the system planning stage. Frequency scanning method [13][14] is a fast way to determine the system resonance frequency. But for each generator, frequency scanning method needs to build equivalent model and then scan the system. Eigenvalue [15][16] and test signal [17][18] methods require detail data information of generator and its shaft model. Besides, the aforementioned 2 methods cost too much computation resource, which is dramatically increasing when the system scale and number of generator increase. Compensation impact factor (CIF) [19] method uses an equivalent source model for generators and only needs the generator capacity and voltage level. Then the CIF for each generator can be determined. With threshold value, the SSR risk can be determined for the generators. The CIF method needs simple calculation process and little information on generator shaft. Therefore, at the planning stage CIF method can be used to determine the short-circuit capacity of each equivalent system, which results in little SSR threat for the generators.
This paper uses the CIF method to swiftly and conveniently determine safety short-circuit capacity group, by which the generators are not under SSR threat. Firstly, CIF is calculated for each generator with different short-circuit capacities of equivalent systems. Then, when the CIFs of all generators are below the threshold value, the short-circuit capacity of each equivalent system is the safety short-circuit capacity. Finally, safety short-circuit capacity group is formed by each safety short-circuit capacity and adopted at the system planning stage to ensure that there is no SSR risk for the generators. The proposed method has two advantages, i.e., 1) simple calculation because of the source model for generator and calculation of CIF, and 2) little information on generator shaft which is not available at the system planning stage.
The remains of this paper are organized as follows, Section II briefly introduces the theory and calculation of CIF. Section III gives the method to obtain safety short-circuit capacity and then forms the group of safety short-circuit capacity. Section IV tests the correctness of the group and Section V draws conclusions of this paper.

II. THE COMPENSATION IMPACT FACTOR
CIF [19] performs the evaluation of the SSR risk caused by fix compensation capacitor for thermal generators. Little information on generator shafts are required to obtain the CIF. As a result, the equivalent model to compute CIF for generators is convenient to build, considering using power source model for the generators. Figure 1 shows active power, which are divided into "causing disturbance group" and "restraining disturbance group", for computing CIF for a generator.
In Figure 1, the generator connects to the system through a compensated line and N non-compensated lines. The "noncompensated line" represents the parallel path to the compensated line. PC is the active power flow in the compensated line while PUC-1 through PUC-N represents the active power flow going through each non-compensated line respectively. The positive directions of these active power flows are shown in Figure 1.
The definition of CIF is shown in (1). Assuming that a fault happens at t1 and then the system recovers to the steady state at t2, if the active power of the causing disturbance group line can be offset perfectly by those of the restraining disturbance group line, then the sub-synchronous disturbance energy has no path to reach the generator end. Hence, there is no interaction energy exchanged between the generator shaft and compensation capacitor, and the generator will not be under any SSR threat. [ , ] 1 However, in practical scenario, CIF is not always equal to 0. But it does not mean that SSR will occur. Therefore, [19] gives a threshold value T CIF =0.2 to forecast that there is no SSR threat for the generator if CIF<T CIF .

A. Equivalent Reactance
In Figure 1, the generator is connected to N+1 AC systems through the compensated line and N uncompensated lines. The equivalent reactance between the generator and the AC system and among the AC systems can simulate short-circuit capacity of each AC system. Assuming that XG-Si and XSi-j are equivalent reactance between the generator and the ith AC system and between the AC system i and j (i, j=1,2,3…N, i ≠ j), an AC system can be equivalent to a vector XSYS, as (2) shows. ( , , ) , where k=1,2,3…m, and m is the number of generators.
With each XSYS, CIFs can be obtained for the generators and the risk of SSR can be determined. If all generators are not suffer SSR risk, then the XSYS are the safety reactance, which represents the short-circuit capacities of the equivalent AC systems.

B. Process to Obtain the Safety Reactance
Suppose the elements of XSYS can form W different combinations which is W XSYS vectors. Figure 2 shows the process to obtain safety reactance XSYS-sty. Inverse the elements of XSYS-sty generates the safety shortcircuit capacities of the AC systems. Then the safety shortcircuit capacity group can be formed by the safety short-circuit capacities.

C. Advantages
Firstly, the proposed method does not cost too much computation resource. The equivalent system model is easy to obtain and the calculation process of CIF is simple. Therefore, compared with other SSR analysis method [13][14][15][16][17][18], the proposed method is more convenient to use.

Advances in Intelligent Systems Research, volume 143
Secondly, the proposed method can be used at system planning stage. (1) shows that the calculation of CIF requires little information on generator shaft. CIF indicates if there is a path for sub-synchronous disturbance energy to excite SSR. Therefore, without the detail data of the generator mode, CIF can be adopted to obtain safety short-circuit capacities of the AC systems. Figure III shows a 3-Bus AC system with a series compensation capacitor line as the test system of this paper. G and T represent a generator and its boost transformer respectively. S 1 and S 2 are two equivalent AC systems which are connected to Bus② and Bus③ with its equivalent reactance Xeq1 and Xeq2. The reactance Xtr represents the electrical strength between Bus② and Bus③. P tr is the active power flowing through X tr . R 1 , X L1 and X C are the resistance, inductive reactance and capacitive reactance of the compensated transmission line, respectively. R 2 and X L2 represent the resistance and reactance of the non-compensated line. P 1 and P 2 are the active power flowing through the compensated and non-compensated transmission lines respectively. X eq1 and X eq2 belongs to X G-Si and X tr is of X Si-j , as (2) shown. The fault at Bus ② adopts 3-phase to ground fault to excite sub-synchronous disturbance component.  (2). When (Xeq1, Xeq2, Xtr) changes as Table 2 listed, X W SYS of Figure 3 is set to start the proposed method. The step of (Xeq1, Xeq2, Xtr) in Table 2 is 0.01 p.u..

B. Safety Reactance
This section calculates CIF with each point of (Xeq1, Xeq2, Xtr) within Table 2. When CIF is smaller than TCIF=0.2 [19], the corresponding (Xeq1, Xeq2, Xtr) point is of the safety reactance XSYS-sty of Figure 2. Figure 4 shows that the CIF with different points of (Xeq1, Xeq2, Xtr). The color bar of each figure shows the value of CIF. The (Xeq1, Xeq2, Xtr) in blue area belongs to XSYS-sty. Figure 4 only gives some fragment surface of the calculation result for convenient observation. The other fragment surfaces show the similar phenomenon and do not presented in this paper.  Figure 4(a) shows that when Xeq2 is fixed, lower values of Xeq1 and Xtr result in a lower CIF value for the generator. Therefore, small reactance between Bus ① and Bus ② and between Bus② and Bus③ assist to avoid SSR risk. When Xeq2 becomes larger, the area of XSYS-sty reduces. Figure 4(b) shows that when Xeq1 is fixed, lower values of Xeq2 and Xtr obtain a lower CIF value for the generator.

Advances in Intelligent Systems Research, volume 143
Therefore, small reactance between Bus ① and Bus ③ and between Bus② and Bus③ are helpful to reduce SSR risk. Compared FIGURE IV(a) with FIGURE IV(b), the area of XSYS-sty is more sensitive with the change of X eq1 rather than the change of X eq2 . Figure 4(c) shows that when X tr is fixed, lower values of X eq1 and X eq2 produce a lower CIF value for the generator. Therefore, small reactance between Bus ① and Bus ② and between Bus① and Bus③ can eliminate the risk of SSR. When X tr increase from 0.2 to 0.6 p.u., the area of XSYS-sty reduces significantly. However, when Xtr continuously increasing, the area of XSYS-sty is not changing obviously.
Therefore, higher short-circuit capacities between the generator and the equivalent AC system and among the equivalent AC systems make contributions to eliminate the risk of SSR.

C. SSR Test
This section selects 3 (X eq1 , X eq2 , X tr ) points to test the correctness of the safety reactance area obtained in section B. Table 3 shows that the CIF values of the 3 (X eq1 , X eq2 , X tr ) points are smaller, near and greater than the threshold value T CIF =0.2. which are located at Zone 1, Zone 2 and Zone 3 respectively. Frequency spectrum analysis, damping characteristics and shaft torque are presented to prove that strong electric strength is critical for the generator to avoid the risk of SSR.  Figure 5 shows SSR test results with the 3 (Xeq1, Xeq2, Xtr) points.  There is no SSR in case 1 because the CIF indicates that the (X eq1 , X eq2 , X tr ) point is of XSYS-sty. Case 2 and 3 show the oscillation phenomenon, especially in case 3 the oscillation is very dramatic. Therefore, the proposed method can determine the safety reactance XSYS-sty and help the system avoid SSR risk.

V. CONCLUSIONS
This paper proposes a method to obtain safety short-circuit capacity group for equivalent AC systems at the system planning stage. CIF is adopted to analysis the risk of SSR instead of analysis of damping characteristics. The equivalent model for CIF and its calculation process are very simple and convenient to achieve. If the CIF of X W SYS is smaller than the threshold value, then X W SYS is of the safety reactance XSYS-sty which should be considered for the planning system due to no SSR risk threating the system. Smaller reactance among the equivalent AC system (higher short-circuit capacity of the equivalent AC system) results in lager area of XSYS-sty and the sub-synchronous disturbance Advances in Intelligent Systems Research, volume 143 energy has no path to be transferred to the generator. Therefore, at the planning stage, smaller reactance of the equivalent system is strongly suggested to avoid the SSR risk.