Nonlinear modal analysis of interaction between torsional modes and SVC controllers

Electric Power Systems Research 91 (2012) 61–70

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Nonlinear modal analysis of interaction between torsional modes and SVC controllers R. Zeinali Davarani ∗ , R. Ghazi, N. Pariz Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

a r t i c l e

i n f o

Article history: Received 14 July 2011 Received in revised form 12 April 2012 Accepted 30 April 2012 Available online 2 June 2012 Keywords: Nonlinear modal analysis Modal series Torsional mode Static VAr compensator

a b s t r a c t This paper deals with the use of Modal Series (MS) method to study the nonlinear interaction between the torsional modes of a turbine-generator set and Static VAr Compensator (SVC) in the stressed power systems. The observations resulting from nonlinear simulations made incentive for suggesting the need for nonlinear analysis of torsional interaction phenomenon in stressed conditions. The obtained results from linear modal analysis were not consistent with those of nonlinear simulations and could not quantify the problem. While the results of nonlinear analysis MS method reveal that, the nonlinear interactions exist between shaft sections and SVC controllers. In the stressed conditions, these nonlinear interactions become more pronounced and it may be a warning for severe damage of shaft segments. By using the indices correspond to the MS method, the nonlinear interactions of torsional modes are evaluated for different conditions. To demonstrate the physical effects of nonlinear interaction of torsional modes with SVC controllers, the torsional torques imposed on shaft sections are provided using the nonlinear simulations. Finally, based on MS method a supplementary controller is designed for SVC to remedy the problem. The proposed approach is implemented on the IEEE 4-machine 11-bus test system. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Static VAr Compensator (SVC) is one of the promising Flexible AC Transmission Systems (FACTS) devices that installed in transmission lines to improve the voltage profile and increase the dynamic and transient stability limits [1]. Due to the rapid growth of the SVC applications in power systems, the problem of its interactions with other system components such as the mechanical system of a turbine-generator set in near the steam power plant, deserves special attention [2]. In the literature, only the linear modal analysis and time-domain simulations have been used to study the interactions between torsional modes and SVC controllers [2–6]. In [2,4], the time-domain simulation is used to study the interactions of torsional modes with SVC. In these references, the damping torque of the synchronous generator is utilized as an index to assess the torsional interaction of the SVC. In [5,6], the interaction is evaluated based on the linearized model of a power system using the eigenvalue analysis. It should be noted that in time domain approaches, the necessary information corresponding to the dynamic behavior of the system is not provided. On the other hand, the linear analysis

∗ Corresponding author. Tel.: +98 511 8763302; fax: +98 511 8763302. E-mail addresses: [email protected], ro [email protected] (R. Zeinali Davarani). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.04.017

methods also cannot demonstrate the nonlinear interaction phenomena [7]. In stressed power systems, especially those equipped with power electronic devices, the nonlinear effects are significantly increased and cannot be neglected. Therefore, in the stressed conditions for precise analyzing the interactions of shaft segments with other system components especially FACTS devices, the more accurate methods are needed. Consequently, an analytical nonlinear modal analysis method should be used to study the torsional interaction phenomenon in stressed conditions. In recent years, the nonlinear modal analysis methods such as the Normal Form (NF), the Modal Series (MS), and the Perturbation Technique (PT) have been used to study the power system dynamic performance [7–17]. These papers have concluded that in the stressed power systems, the possibility of the nonlinear interactions is increased, and thereby the performance of power system controllers will deteriorate. As shown in [14,18], the MS method exhibits some advantages over the normal form method. Its validity region is independent of the modes resonance, does not require nonlinear transformation, and it can be easily applied on large power systems. Therefore, in the present paper for the first time, the MS method is used to study the nonlinear interactions between torsional modes of shaft segments and the FACTS controllers in the stressed conditions. For this purpose, the dynamic equations of the power system equipped with SVC are provided. By performing the firstorder Taylor expansion of these equations and using the linear

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modal analysis method, the eigenvalues and the corresponding system components are determined. Then, by considering the first and second-order Taylor expansion of equations, the second-order approximate solution of MS method is obtained. Utilizing the MS based nonlinear interaction indices, the nonlinear interaction of torsional modes are evaluated for different conditions and the most interacting factors are determined. To demonstrate the physical effects of nonlinear interaction on torsional modes, the torsional torques imposed on the turbine-generator shafts are also evaluated through simulations. Finally, regarding the interacting factors a supplementary controller is designed for SVC to attenuate the shaft stresses resulting from nonlinear interactions with SVC controllers. The proposed approach is applied on the IEEE 4-machine 11-bus test system. This paper is organized as follows. In Section 2, the torsional interaction with FACTS devices is described. The required steps in performing the nonlinear modal analysis of torsional interactions are introduced in Section 3. In Section 4, the case study and the obtained results are provided. The necessary discussions and remarks are also appeared in this section. The conclusions are given in Section 5. 2. Torsional interaction phenomenon in power systems 2.1. Basic concept A steam power plant turbine-generator set consists of several masses connected to each other by shafts having bounded stiffness coefficients. The modes associated with these masses are known as torsional modes. The SVC installation in power systems will add new modes to systems, so make them susceptible to interact with torsional modes. Depending on the power system complexity, conditions and the SVC control parameters, the torsional interaction may be negligible or critical. In the critical case, in physical terms, the amplitudes of torsional oscillations and their periods become quite great which may cause a significant reduction in the lifetime of the shaft in the oscillations periods. 2.2. Modal analysis of torsional interaction By modal analysis the valuable information corresponding to the stability of torsional modes and their interactions with other system modes are obtained. The interaction of torsional modes occurs in different ways and can be classified as linear or nonlinear. The linear and nonlinear modal analysis methods can be used to study such interactions. 2.2.1. Linear modal analysis of torsional interaction In linear torsional interactions, the torsional modes are affected by the natural modes of other components such as SVC controllers [1]. Hence, the torsional modes experience a lower level of damping. To study this type of interaction the linear modal analysis method is applied and the effects of various parameters on the damping of torsional modes are investigated. As this approach cannot identify the nonlinear interactions, the obtained results may not be reliable when the system encounters nonlinearities resulting from stressed conditions and/or using FACTS controllers. 2.2.2. Nonlinear modal analysis of torsional interaction In nonlinear torsional interactions, the nonlinear effects lead to the appearance of new modes known as combination modes or nonlinear modes in the time solution of torsional modes. In addition, the nonlinearity contribution factors will affect the amplitude of torsional modes. For instance, in responses associated with torsional modes the pronounced presence of nonlinear modes related to the SVC modes indicate the nonlinear interaction of torsional

modes with SVC controllers. When this nonlinear interaction is increased, a serious warning is provided regarding the excessive torsional torques imposed on the turbine-generator shafts. In addition, in this condition the torsional frequency analysis obtained from linear modal analysis is not reliable. As in stressed power systems, especially those equipped with FACTS devices, the results obtained from linear modal analysis are not reliable; the nonlinear modal analysis should be used to study the torsional interaction phenomenon. To do so, several methods such as NF, MS, and PT methods can be employed. With respect to the advantageous of MS method over others [14,18], this method is utilized to study the nonlinear torsional interaction. The mathematical equations of MS method are provided in Appendix A (Eqs. (A.1)–(A.7)). In order to evaluate the effects of nonlinearity on the torsional interactions, the proper nonlinear interaction indices should be used. Based on the MS method, the MI1 and MI2 indices are defined by (1) and (2) as follows. These indices determine the levels of nonlinear interaction of system modes with torsional modes.

⎛ MI1(j) = ⎝

 

|yj0 | − yj0 −

n n k=1

⎞  ⎠

j

h2kl yk0 yl0  l=1

yj0

  j   max(h2kl yk0 yl0 )   k,l  MI2(j) =  n n  j  yj0 − k=1 l=1 h2kl yk0 yl0 

(1)

(2) (k,l,j)/ ∈R

2

The index MI1 determines that the torsional modes how and how much are affected by the nonlinear effects. For instance, the negative high value of index MI1 at torsional mode j shows that the nonlinear effects will increase the amplitude of this torsional mode and its frequency becomes more important. The index MI2 illustrates that which combination modes and the corresponding system components have the most contribution in the nonlinear torsional interactions. 3. Approach of nonlinear modal analysis In order to perform the nonlinear modal analysis of torsional interactions based on the MS method, the following steps to be taken. (i) The differential dynamic equations governing the studied power system are provided. (ii) The stable equilibrium point (Xsep ) is obtained by power flow solution of test power system and solving the algebraic equations of system components. (iii) The natural eigenvalues of the power system are obtained using the first-order Taylor expansion of system equations around the stable equilibrium point as determined in step II. (iv) By linear modal analysis, the participation matrix is obtained and then the torsional, SVC, and other system modes are remarked. (v) Regarding the first and second-order Taylor expansion of power system equations, the second-order approximate solution of MS method is obtained. (vi) The MS based nonlinear interaction indices at torsional modes are calculated and the levels of nonlinear interactions are determined. Then those system components that have the most contribution in nonlinear interaction of torsional modes are specified. In addition, the new frequencies that might appear in the responses related to torsional modes are identified.

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70

G1

1

5

Area 1

7

6 25KM

10KM

2

8 55KM

9 55KM

11

3

G3

25KM

C9 L9

L7 C7

G2

10 10KM

63

4

Area 2

G4

SVC Fig. 1. Single line diagram of the test system.

Fig. 2. SVC control system block diagram.

(vii) The steps I to VI are repeated for different conditions and the most effective interacting factors in nonlinear interaction of torsional modes are determined. (viii) To observe the physical effects of the nonlinear interaction of torsional modes, the torsional torques imposed on the shaft segments of the turbine-generator set are provided by implementing the nonlinear simulations. 4. Case study For analyzing the nonlinear interactions between steam power plant turbine-generator set and SVC controllers, a two-area fourmachine power system equipped with SVC has been used [20]. Its single-line diagram and the block diagram of the SVC controller are shown in Figs. 1 and 2 respectively. The system parameters are found in [20,21], and their summaries are listed in Appendix A. Each turbine-generator set is modeled by five masses as shown in Fig. 3. In this case, each machine has four torsional modes. The differential equations describing the turbine, synchronous generators and SVC are presented in Appendix A (Eqs. (A.8)–(A.11)).

GEN: Generator Rotor LP: Low Pressure Turbine Section HP: High Pressure Turbine Section Fig. 3. The model of turbine-generators in the test system.

4.1. Eigenvalue analysis of studied system For each generator, 3 electrical and 10 mechanical equations are considered; by including four equations for SVC, the total of 56 equations are obtained. The state variables of the test system are: E  qi , Ed , Efdi , ıGENi , ωGENi , ıLPAi , ωLPAi , ıLPBi , ωLPBi , i

ıLPCi , ωLPCi , ıHPi , ωHPi , Vmeas , V1 , Bref , Bsvc

(3)

The fundamental modes and the corresponding system components are determined for the normal condition using the approximate linear analysis. Some of these modes are shown in Table 1. As seen from Table 1, the test system has 16 torsional and 4 SVC modes. 4.2. Nonlinear modal analysis of interaction between turbine-generator shafts and other system components To evaluate the nonlinear interaction of the steam turbinegenerators, for normal and stressed conditions the MS nonlinear interaction indices are calculated at torsional modes. The obtained results are reported in Table 2. The stressed condition is provided by increasing the amount of power transmitted from area 1 to area 2. This is accomplished via varying the amounts of loads in buses 7 and 9. In normal condition, the amounts of loads are such that the transmitted power is equal to 200 MW. In stressed conditions, this power is increased to 400 MW via varying the loads. The results of full load power flow for normal and stressed conditions are reported in Appendix A (Tables A.1 and A.2). Referring to Table 2, the following remarks are made: • In the presence of SVC, the maximum nonlinear interaction of torsional modes occurs with SVC controllers (as seen from Table 1,

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Table 1 Eigenvalue of the test system in normal condition. Mode

Eigenvalue

Frequency (Hz)

Source

Mode

Eigenvalue

Frequency (Hz)

Source

1 2, 3 4, 5 6, 7 8, 9 10, 11 12, 13 14, 15 16, 17 18, 19 24, 25

−521 −2.72 ± j221.2 −0.3 ± j148.4 −0.3 ± j148.4 −0.27 ± j141.5 −0.27 ± j141.5 −0.04 ± j127.5 −0.03 ± j127.1 −0.04 ± j121.2 −0.04 ± j121.6 −0.06 ± j95.8

0 35.2 23.6 23.6 22.5 22.5 20.3 20.2 19.3 19.4 15.3

SVC SVC Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2)

26, 27 28, 29 30, 31 32, 33 34, 35 36, 37 38, 39 40 43, 44 45, 46 47, 48

−0.05 ± j95.5 −0.05 ± j91 −0.05 ± j91.4 −0.06 ± j52.7 −0.04 ± j51.9 −0.05 ± j49.4 −0.06 ± j50.2 −33.11 −0.56 ± j7.3 −0.56 ± j7 −0.02 ± j4.2

15.2 14.5 14.5 8.4 8.3 7.9 8 0 1.2 1.1 0.7

Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) SVC Local(G1,G2) Local(G3,G4) Inter area(G1,G2,G3,G4)

Table 2 Nonlinear interaction indices at torsional modes in different conditions. Torsional mode

Normal condition

Stressed condition

MI1

MI2

MI1

MI2

6 (G1) 14 (G1) 26 (G1) 34 (G1)

0.06 0.05 0.01 −0.09

0.05(2,3) 0.05(2,3) 0.03(2,3) 0.06(2,3)

0.26 0.27 0.12 −0.65

0.21(2,3) 0.23(2,3) 0.14(2,3) 0.28(2,3)

4 (G2) 12 (G2) 24 (G2) 32 (G2)

−0.03 −0.07 −0.04 −0.12

0.04(2,3) 0.04(2,3) 0.02(2,3) 0.11(2,3)

0.13 −0.05 −0.16 −1.68

0.18(2,3) 0.18(2,3) 0.13(2,3) 0.36(2,3)

8 (G3) 16 (G3) 28 (G3) 36 (G3)

0.19 0.13 0 0.1

0.12(2,3) 0.07(2,3) 0.12(2,3) 0.06(2,3)

0.06 0.2 −0.21 0.42

0.39(2,3) 0.3(2,3) 0.33(2,3) 0.38(2,3)

10 (G4) 18 (G4) 30 (G4) 38 (G4)

0.26 0.18 0.02 0.13

0.10(2,3) 0.05(2,3) 0.09(2,3) 0.07(2,3)

−0.02 0.23 −0.16 0.51

0.21(2,3) 0.22(2,3) 0.24(2,3) 0.49(2,3)

The bold values show that in those situations the interaction phenomenon is more severe.

•

•

•

For further investigation of nonlinear interactions and to characterize the interacting factors, the following several cases are studied. 4.2.1. Effect of stressed condition on the modal analysis of the torsional interaction Fig. 4 depicts the root-loci of the important modes of our test system where the transmitted power increases from 200 MW (normal condition) to 400 MW (stressed condition). By tracing the variation

Imaginary Part of Eigenvalue (rad/s)

•

modes 2 and 3 correspond to SVC) and greatly increases in the stressed conditions. By comparing the values of MI1 and MI2 indices corresponding to G1 and G2 and also for G3 and G4, the following interesting results are obtained. It is observed that in each area, the turbinegenerator set which is electrically closer to the SVC, receives a greater nonlinear interaction than the others. For instance, in the stressed condition, the maximum values of index MI1 at torsional modes of G1 and G2 are equal to −0.65 and −1.68 respectively. In addition, at the torsional modes of generators G3 and G4, the maximum values of index MI1 is 0.42 and 0.51 respectively. The nonlinear interaction between torsional modes of a given generator with SVC controllers is not at the same level. For instance, in this test system, the torsional mode having lower frequency experiences the greater level of nonlinear interaction. As seen from Table 1, the SVC has one swing mode (2 and 3) and two real modes (1 and 40). The index MI2 reveals that the maximum interaction of torsional modes occurs with the combination mode (2, 3), which arises from SVC swing mode. As this combination mode (2 + 3 ) has only the real value, the new important frequency is not appeared in the output system response pertaining to torsional modes. In the stressed condition, the high negative value of index MI1 at torsional modes 32 and 34 indicates that the nonlinear effects will severely excite these torsional modes, so their amplitudes are greatly increased and their frequencies become more important. As the linear analysis cannot provide the similar results, the FFT analysis provided by the approximate linear method cannot be acceptable.

250

200

SVC Mode

Torsional Modes

160 140

150

120 100 80

100

60 40 -0.3

-0.25

-0.2

-0.15

-0.1

-0.05

50 Local Modes

0 -3

-2.5

-2

-1.5

-1

0

Inter-area Mode

-0.5

0

Real Part of Eigenvalue(1/s) Fig. 4. Root-loci of the test system eigenvalue where the transmitted power increased.

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70 0.5

Index MI2 at torsional modes

Index MI1 at torsional modes

1 0.5 0 -0.5 -1 -1.5 -2 200

65

Mode 32 (G2) Mode 34 (G1) Mode 36 (G3) Mode 38 (G4)

240

280

320

360

400

Mode 32 (G2) Mode 34 (G1) Mode 36 (G3)

0.4

Mode 38 (G4)

0.3 0.2 0.1 0 200

Transmitted Power (MW)

240

280

320

360

400

Transmitted Power (MW)

Fig. 5. Nonlinear interaction indices at torsional modes at various transmitted power.

0.4

0.4 Transmitted Power=200MW

Transmitted Power=200MW

Transmitted Power=300MW Transmitted Power=400MW

0.3 0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2 0.35

0.45

0.55

0.65

Transmitted Power=300MW Transmitted Power=400MW

0.3

0.75

-0.2 0.35

0.45

0.55

0.65

0.75

0.2

0.2 Transmitted Power=200MW

Transmitted Power=200MW

Transmitted Power=300MW Transmitted Power=400MW

Transmitted Power=300MW Transmitted Power=400MW

0.1

0.1

0

0

-0.1

-0.1

-0.2 0.35

0.45

0.55

0.65

0.75

-0.2 0.35

0.45

0.55

0.65

0.75

Fig. 6. The torsional torque imposed on the turbine-generator at various transmitted power.

of modes damping, we can see that the damping of SVC swing mode is significantly reduced while the damping of torsional modes is not affected. Therefore, the linear analysis method shows that the torsional modes are not severely excited.

To study the effect of stressed condition on the nonlinear modal analysis of the torsional interaction, the indices MI1 and MI2 are calculated at torsional modes for different amounts of transmitted power (i.e. for different levels of stress). The results for some

0.1

0.1

Normal Condition Stressed Condition

0.05

0.05

0

0

-0.05

-0.05

-0.1

-0.1 Normal Condition Stressed Condition

-0.15

-0.15

0.35

0.45

0.55

0.65

0.75

0.35

0.45

0.55

0.65

Fig. 7. The torsional torque imposed on the turbine-generator at different conditions when SVC is not in operation.

0.75

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Table 3 Nonlinear interaction indices at torsional modes in stressed condition when SVC is not in operation. Torsional mode

MI1

MI2

Torsional mode

6 (G1) 14 (G1) 26 (G1) 34 (G1)

0.07 0.12 0.07 0.08

0.05(47,47) 0.15(47,47) 0.05(47,47) 0.12(47,47)

8 (G3) 16 (G3) 28 (G3) 36 (G3)

MI1 0.06 0.08 0.05 0.14

MI2 0.06(47,47) 0.03(47,47) 0.09(47,47) 0.04(47,47)

4 (G2) 12 (G2) 24 (G2) 32 (G2)

0.06 0.08 0.05 0.14

0.03(47,48) 0.07(47,48) 0.03(47,48) 0.11(47,47)

10 (G4) 18 (G4) 30 (G4) 38 (G4)

−0.03 −0.01 −0.03 −0.03

0.02(47,47) 0.01(47,48) 0.02(47,48) 0.02(47,47)

The bold values show that in those situations the interaction phenomenon is more severe.

torsional modes that experienced higher nonlinear interaction are shown in Fig. 5. As the transmitted power increases, the nonlinear interaction index MI2 is almost increased for all torsional modes. However, the increase rate of index MI2 at torsional modes 36 and 38 is greater than the other modes. Furthermore, the variation of index MI1 shows that the increase of transmitted power will greatly affect some torsional modes such as modes 32 and 34. It can be seen that in the stressed conditions, the obtained results from linear and MS methods are not identical. The linear analysis does not reveal the significant interaction of torsional modes with other system components. While the MS method shows that, a severe nonlinear interaction exists between torsional modes and SVC controllers. 4.2.2. Physical effects of nonlinear torsional interaction To study the physical effects of nonlinear torsional interaction on the performance of the turbine-generator, the torsional oscillations are calculated using the step-by-step solution of nonlinear equations. For various amounts of transmitted power, the torsional torques which impose on the turbine-generator shafts are shown in Fig. 6. It is seen that as the amount of transmitted power increases the amplitudes of the torsional oscillations will substantially increase. By comparing Figs. 5 and 6, it is found that there is an exact consensus between the increase of nonlinear interactions and the increase of torsional oscillations. Therefore, in stressed power systems, the nonlinear modal analysis based on MS method can

well predict the destructive effects of nonlinearity on the shaft segments. 4.2.3. Evaluate the nonlinear interactions of torsional modes in the absence of SVC It is interesting to investigate the possibility of occurrence of nonlinear interactions of torsional modes with other modes in the absence of SVC. For this case, the nonlinear interaction of torsional modes is studied under stressed conditions, and the results are shown in Table 3. It is seen that when SVC is not in operation, the maximum of nonlinear interactions of shaft segments will take place with inter-area mode (as seen from Table 1, the modes 47 and 48 reveal inter-area mode). However, the amount of interactions cannot be a major concern even in the stressed condition. In the absence of SVC the torsional torques imposed on the turbine-generator shaft are shown in Fig. 7 for normal and stressed conditions. As predicted, in this case the amplitudes of torsional oscillations are considerably reduced in compare with those obtained in the existence of SVC. 4.2.4. SVC supplementary controller design to reduce the nonlinear torsional interaction According to the obtained analytical results and the quite important matter of shaft vulnerability, the nonlinear interactions of torsional modes should be studied carefully. Then to protect the shaft segments from any undesirable nonlinear interactions effects,

Table 4 Effect of SVC supplementary controller on the nonlinear interaction of torsional modes. Torsional Modes

Nonlinear interaction index MI2 Without NIRSC

With NIRSC Case 1 T1N = 0.4 T2N = 0.03 KN = 0.033

Case 2 T1N = 0.4 T2N = 0.06 KN = 0.033

Case 3 T1N = 0.4 T2N = 0.03 KN = 0.066

Case 4 T1N = 0.4 T2N = 0.03 KN = 0.033

Case 5 T1N = 0.4 T2N = 0.06 KN = 0.033

Case6 T1N = 0.4 T2N = 0.03 KN = 0.066

6 (G1) 14 (G1) 26 (G1) 34 (G1)

0.28 0.31 0.18 0.33

0.25 0.32 0.17 0.21

1.29 1.02 0.44 0.40

0.22 0.20 0.17 0.20

0.20 0.57 0.04 0.05

0.17 0.17 0.09 0.20

0.04 0.05 0.02 0.02

4 (G2) 12 (G2) 24 (G2) 32 (G2)

0.23 0.25 0.17 0.41

0.21 0.29 0.17 0.32

1.01 0.78 0.41 0.49

0.25 0.32 0.26 0.38

0.10 0.17 0.03 0.07

0.14 0.14 0.08 0.27

0.04 0.04 0.01 0.03

8 (G3) 16 (G3) 28 (G3) 36 (G3)

0.41 0.35 0.36 0.56

0.37 0.24 0.37 0.26

0.47 0.46 0.53 0.79

0.20 0.17 0.23 0.16

0.23 0.07 0.13 0.04

0.38 0.21 0.25 0.15

0.11 0.04 0.07 0.02

10 (G4) 18 (G4) 30 (G4) 38 (G4)

0.23 0.24 0.26 0.69

0.16 0.14 0.24 0.30

0.20 0.23 0.32 0.60

0.06 0.06 0.09 0.12

0.08 0.04 0.08 0.04

0.30 0.16 0.16 0.17

0.06 0.02 0.03 0.02

The bold values show that in those situations the interaction phenomenon is more severe.

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70

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Table 5 Effect of SVC supplementary controller on the damping of system modes. Mode number

Source

Real part of system modes Without NIRSC

2, 3 4, 5 6, 7 8, 9 10, 11 12, 13 14, 15 16, 17 18, 19 24, 25 26, 27 28, 29 30, 31 32, 33 34, 35 36, 37 38, 39 43, 44 45, 46 47, 48

SVC Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Torsional(G2) Torsional(G1) Torsional(G3) Torsional(G4) Local(G1,G2) Local(G3,G4) Inter area(G1,G2,G3,G4)

sTw 1+ sTw

−0.618 −0.299 −0.299 −0.272 −0.272 −0.039 −0.019 −0.044 −0.036 −0.056 −0.040 −0.054 −0.051 −0.061 −0.030 −0.049 −0.056 −0.560 −0.543 −0.008

Case 1 T1N = 0.4 T2N = 0.03 KN = 0.033

Case 2 T1N = 0.4 T2N = 0.06 KN = 0.033

Case 3 T1N = 0.4 T2N = 0.03 KN = 0.066

Case 4 T1N = 0.4 T2N = 0.03 KN = 0.033

Case 5 T1N = 0.4 T2N = 0.06 KN = 0.033

Case 6 T1N = 0.4 T2N = 0.03 KN = 0.066

−1.872 −0.299 −0.299 −0.272 −0.272 −0.039 −0.014 −0.048 −0.036 −0.055 −0.035 −0.058 −0.051 −0.061 −0.018 −0.059 −0.056 −0.564 −0.541 −0.016

−1.259 −0.299 −0.299 −0.272 −0.272 −0.039 −0.016 −0.046 −0.036 −0.055 −0.037 −0.056 −0.051 −0.061 −0.022 −0.056 −0.056 −0.564 −0.540 −0.009

−3.137 −0.299 −0.299 −0.272 −0.272 −0.039 −0.009 −0.052 −0.036 −0.055 −0.030 −0.062 −0.051 −0.060 −0.005 −0.069 −0.056 −0.568 −0.538 −0.025

−3.261 −0.299 −0.299 −0.272 −0.272 −0.039 −0.014 −0.048 −0.036 −0.056 −0.035 −0.058 −0.051 −0.061 −0.021 −0.057 −0.056 −0.560 −0.538 −0.018

−2.142 −0.299 −0.299 −0.272 −0.272 −0.039 −0.016 −0.046 −0.036 −0.056 −0.037 −0.056 −0.051 −0.061 −0.024 −0.054 −0.056 −0.560 −0.537 −0.011

−6.335 −0.299 −0.299 −0.272 −0.272 −0.039 −0.007 −0.053 −0.037 −0.056 −0.030 −0.062 −0.052 −0.061 −0.007 −0.067 −0.057 −0.559 −0.525 −0.057

1+ sT3 N 1+ sT4 N

1+ sT1N 1+ sT2N

KN

With NIRSC

Vscs

Fig. 8. SVC supplementary controller block diagram.

a suitable technique should be used to reduce them as far as possible. One procedure is to design a nonlinear interaction reduction supplementary controller (NIRSC) for SVC to reduce the nonlinear torsional interaction with SVC controllers. For this purpose, the appropriate supplementary input signals should be selected [22]. As the SVC controller and the level of tie line power flow in the test system are the main interacting factors in the nonlinear interaction of torsional modes, the reference susceptance Bref in SVC controller (Fig. 2) and the line power PL are considered as the input signals. The block diagram of such supplementary controller for SVC is shown in Fig. 8, which is similar to that defined in [23]. The NIRSC should be designed such that in the worst condition of stress the nonlinear torsional interaction to be reduced as much as

possible. In the test system, the worst stressed condition is defined by transmitted power of 420 MW. In this condition, the parameters of NIRSC are obtained by trial and error method. Regard to maintaining the stability of system modes, the reduction of nonlinear torsional interaction is considered as the objective function. For different parameters of NIRSC, the levels of nonlinear interaction of torsional modes are presented in Table 4 as cases (1–6). In addition, for these cases the damping of torsional and dominant modes is presented in Table 5. In cases 1–3, the PL and in cases 4–6 the Bref are used as input signals for NIRSC. In all cases, the parameters Tw , T3N , and T4N , respectively are equal to 1, 0, and 0.033 s. The best parameters for NIRSC are obtained by trial and error method as shown in case 4. As seen from Table 4, in cases 4 and 6, the NIRSC noticeably decreases the level of nonlinear torsional interaction. However, Table 5 shows that in case 6 the damping of some torsional modes is decreased. The torsional torques imposed on the turbinegenerator shafts for case 4 are shown in Fig. 9. It is seen that by using the NIRSC, the amplitudes of torsional oscillations are

0.4

0.4

Without NIRSC

Without NIRSC

With NIRSC

With NIRSC

0.3

0.3

0.2

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2 0.35

0.45

0.55

0.65

0.75

-0.2 0.35

0.45

0.55

Fig. 9. Effect of NIRSC on the torsional torques imposed to the turbine-generators shaft.

0.65

0.75

68

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70

greatly reduced so there is no risk of shafts damage from nonlinear interactions.

solution of (A.1) is obtained as follows [14]:

⎛

⎝yj0 −

yj (t) =

5. Conclusion

+

n n 

+

X˙ = F(X)

j

Ckl yk0 yl0



(k,l,j)/ ∈R

2

tej t

(A.6) (k,l,j)/ ∈R

2

j Ckl /(k

R2 = {(k, l, j)|k + l − j | < 0.001|j |}

(A.7)

A.2. Algebraic equations of power system A.2.1. Synchronous generator equations The simple d-q axes model is used to represent the synchronous generator and the thyristor-based exciter having a high transient gain. Efdi − Eq i + (xdi − xd )Iqi

E˙ q i =

i

 Td0

i

−Ed − (xqi − xq i )Idi i

(A.8)

 Tq0

i

E˙ fdi =

The MS method is briefly described as follows. The detailed mathematical equations of this method are found in [14]. If (A.1) exhibits the nonlinear dynamic equations of a power system with N variables, (A.2) will be the Taylor expansion of (A.2) around a stable equilibrium point Xsep .



= + l − j ), and R2 refers to a set of eigenvalues leading to second order quasi resonances and defined as:

Appendix A. A.1. Modal series method



k=1 l=1

i

1 (−Efdi + Kexci (Vrefi − Vti + Vsi )) Texci

where the state variables and parameters have the same meaning as in [19]. Vsi is supplementary signal of the excitation system and Vti is the terminal voltage of the ith machine and obtained as: Vti =

(Eq i + xd Idi )2 + (Ed − xq i Iqi )2 i

i

(A.9)

(A.1) 1 t i X H X + ··· 2

i = 1, 2, . . . , N

(A.2)

In (A.2) A and H, respectively, denote the Jacobian and Hessian matrices. By applying the linear transformation X = UY to first and second order terms of (A.2), yields: y˙ j = j yj + Y t CY = j yj +

n n  

j

Ckl yk yl

j = 1, 2, . . . , n

(A.3)

k=1 l=1

where j is eigenvalue, U and V, respectively, denoting the matrices of right and left eigenvectors of Jacobian matrix A, and C is defined as: 1 t Vjp (U t H p U) j = 1, 2, . . . , n 2 n

A.2.2. Mechanical equations The steam turbine-generator set is represented by a five-mass model and for each mass two equations are considered as [20]: Mij ω˙ ij = Tij − Dij ωij + Kij−1,j (ıij−1 − ıij ) − Kij,j+1 (ıij − ıij+1 ) ı˙ ij = ωb ωij

A.2.3. SVC control system equations With reference to block diagram of SVC controller shown in Fig. 2, the SVC control system equations obtained as: V˙ meask =

If in (A.3) only the first term is considered, and the higher-order terms are neglected, the linear approximate solution of (A.1) in Ycoordinate becomes [14]: (A.5)

In addition, in (A.3) by considering the first and second order terms and using MS method, the second-order approximate

(A.10)

where Mij , Tij and Dij , respectively, denote inertia constant, mechanical torque and damping of jth mass of ith machine, Kij,j+1 is the stiffness between two masses j and j + 1, and mi is the number of turbine-generator components of the ith machine.

(A.4)

p=1

j t yj (t) = yj0

(k,l,j)/ ∈R 2

j h2kl yk0 yl0 e(k +l )t

 n n 

j where h2kl

⎞ ⎠ ej t

j

h2kl yk0 yl0

k=1 l=1

E˙ d =

Cj =



k=1 l=1

The results of this paper show that in the stressed power systems, there exist a nonlinear interaction between turbine generator shaft and other system components. In the presence of Static VAr Compensator (SVC), the nonlinear interactions occur with SVC controllers. The results show with increasing the degree of stress in the power system, the nonlinear interaction between torsional modes and SVC controllers significantly increased. Although under stressed conditions, the damping of torsional modes may not be affected but their amplitudes can be considerably influenced. In addition, the nonlinear effects led to the appearance of combination modes in the responses related to torsional modes. In such cases, the linear analysis method cannot detect the increased torsional oscillations. While the nonlinear modal analysis based on the Modal Series method, well shows the intensive nonlinear interaction between torsional modes and SVC controllers. Simulation results show that the physical effects of the nonlinear interaction of torsional modes appear as excessive torques imposed on the shaft segments. Furthermore, the results show that depending on the area characteristics and the electrical distance of SVC, the nonlinear interaction can be negligible or critical. In addition, a nonlinear interaction reduction supplementary controller (NIRSC) for SVC is developed to reduce the nonlinear interaction between torsional modes and SVC controllers.

x˙ i = Ai X +

n n 

V˙ 1k =

Vtsvck − Vmeask Tmk

VREFk + Vscsk − Vmeask − T1k V˙ meask − V1k

B˙ refk = B˙ svck =

KRk V1k − Brefk

T2k

TRk Brefk + Bscsk − Tdk B˙ refk − Bsvck Tbk

(A.11)

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70

where Vscsk and Bscsk are supplementary control signals and Vtscsk is the kth SVC terminal voltage.

A.3.2. Mechanical data Mass

Inertia constant of G1 and G2

Inertia constant of G3 and G4

Shaft segment

Spring constant (pu Torque/rad)

HP LPC LPB LPA GEN

0.176 1.427 1.428 1.428 0.869

0.194 1.569 1.571 1.571 0.956

HP-LPC LPC-LPB LPB-LPA LPA-GEN

17.78 27.66 31.31 37.25

A.3. Test system data All parameters are in per unit except time and inertia constant. A.3.1. Synchronous generator data (Vbase = 20 kV, Sbase = 900 MVA) Xd = 1.8, Xq = 1.7, Xd = 0.3, Xq = 0.55  Td0

=

 8s, Tq0

69

A.3.3. Transmission network data (Vbase = 230 kV, Sbase = 100 MVA)

= 0.4, Ra = 0.0025

PL7 = 1167 MW, QL7 = 100 MVAr

RL = 0.0001 pu/km, XL = 0.0001 pu/km, bc = 0.0001 pu/km

PL9 = 1567 MW, QL9 = 100 MVAr A.3.4. SVC control system data

QC7 = 1167 MW, QC7 = 100 MVAr

KR = 60, TR = 0.1 s, T1 = 0.03 s,

In case of without SVC

T2 = 0.02 s, Tb = 0.004 s, Td = 0.001 s, Tm = 0.004 s,

QC7 = 200 MVAr, QC9 = 350 MVAr

Qsvc = ±350 MVAr A.3.5. Power flow solution of test system

Table A.1 Load flow solution for the normal (No.) and stressed (St.) conditions (Sb = 100 MVA). Bus number

Voltage (pu)

Generation (pu) Active

1 2 3 4 5 6 7 8 9 10 11

Load (pu) Reactive

SVC generation (pu)

Active

Reactive

No.

St.

No.

St.

No.

St.

No.

St.

No.

St.

No.

St.

1.030 1.010 1.030 1.010 1.007 0.980 0.963 1.000 0.973 0.985 1.009

1.030 1.010 1.030 1.010 1.006 0.978 0.961 1.000 0.963 0.979 1.006

7.00 7.00 7.038 7.00 – – – – – – –

7.00 7.00 7.189 7.00 – – – – – – –

1.815 2.262 1.686 1.929 – – – – – – –

1.856 2.36 1.883 2.317 – – – – – – –

– – – – – – 11.67 – 15.67 – –

– – – – – – 9.67 – 17.67 – –

– – – – – – 1 – 1 – –

– – – – – – 1 – 1 – –

– – – – – – – 0.992 – – –

– – – – – – – 1.914 – – –

Table A.2 Power flow solution for the normal and stressed conditions (Sb = 100 MVA). Line number

1 2 3 4 5 6 7 8 9 10 11 12

From bus

1 2 3 4 5 6 7 7 8 8 9 10

To bus

5 6 11 10 6 7 8 8 9 9 10 11

Power injection in normal condition (pu)

Power injection in stressed condition (pu)

Active

Reactive

Active

Reactive

7 7 7.038 7 7 13.877 1.002 1.002 0.989 0.989 −13.715 −6.915

1.815 2.262 1.686 1.929 0.994 1.182 −0.448 −0.448 0.01 0.01 1.246 0.329

7 7 7.189 7 7 13.877 2.002 2.002 1.953 1.953 −13.851 −7.059

1.856 2.36 1.883 2.317 1.032 1.306 −0.393 −0.393 0.261 0.261 0.878 0.245

70

R. Zeinali Davarani et al. / Electric Power Systems Research 91 (2012) 61–70

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