Abstract-The dynamic modal interaction between the power system modes due to the variation of system parameters can cause strong resonance. The effect of strong resonance on these oscillatory modes can sometimes be a concern for the system stability. This paper investigates the performance of these modes, the swing mode (SM) and the exciter mode (EM), in the presence of a Static Synchronous Series Compensator (SSSC) damping controller. The significant modes of interest are identified and separated based on the methods of participation factors and multi-modal decomposition. Strong resonance is observed in full order system and reduced order system, which is based on the method of perturbations, suggested by Seiranyan. The systems of study include the 3-machine 10-bus system, 4-machine 2-area system and a practical system with 4-machine 11 bus- a part of the southern grid- an Indian power system.
Keywords-Dynamic modal interaction, strong resonance, oscillatory modes, method of perturbations, modal decomposition, supplementary modulation controller
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Introduction
The power system stability can be evaluated by an analysis of its modes as a consequence of the system linearization. Interaction or coupling between the oscillatory modes occurs when one or more system parameters are varied. This could be due to the equivalence between eigenvectors and eigenvalues [1], [2], [6]. The modes that are located at an extremity move closer to each other and can lead to collision in such a way that one of the modes may subsequently become unstable. Strong resonance is differentiated by the concurrence of eigenvectors and eigenvalues, and any system input along one eigenvector will excite the other mode.
The eigenvalue sensitivity analysis frequently observed in oscillatory systems during parametric studies is carried out by Seyranian [5], [12]. The performance of eigenvalues of autonomous oscillatory systems depending upon multiple parameters is analyzed in the neighborhood of a multiple point. Sensitivity analysis based on the perturbation method of eigenvalues and eigenvectors, illustrates the interaction of eigenvalues by a family of hyperbolae.
Dobson [6], [7] have studied the interaction of oscillatory electric power system modes due to possible perturbations near strong resonance. Padiyar and Saikumar have analyzed strong resonance [1], [2] due to the variation of controller parameters associated with a STATCOM supplementary modulation controller [15], [16], [17]. The interaction of eigenvalues near the multiple points based on the theory suggested by Seiranyan is applied to systems with feedback controller. Nomikos and Vournas [8] have analyzed the strong resonance among electromechanical and exciter modes in multimachine systems and their effects on the design of PSS.
Analysis of dynamic modal interactions is essential to alleviate them and hence a high performance power system can be achieved which can operate reliably within secured limits. An important step in developing improved methods of operating the power system consists of illustrating mechanisms, which cause oscillations. Therefore, the present paper extensively evaluates the presence of strong resonance associated with a SSSC. Interaction of eigenvalues between exciter mode (EM) and swing mode (SM) is noticed initially while tuning the SSSC parameters in the full order system. Applying the concept of multi modal decomposition [11] the swing mode that interacts with the exciter mode is isolated and is then identified by computing participation factors. The system size is reduced, by retaining only the modes of interest. The asymptotic behavior of the two modes in the neighborhood of strong resonance is analyzed by extending the theory suggested by Seiranyan [4], [5] and compared with the root loci of the systems. Results obtained in both the test and practical systems illustrate the presence of strong resonance in full order and reduced order systems.
Supplementary modulation controller for SSSC
SSSC a voltage-sourced converter injects a voltage in series with the line. The magnitude of the injected voltage can be controlled independently of the line current for the purpose of increasing or decreasing the overall reactive voltage drop across the line. Fig. 1 shows the single line diagram of SSSC connected in series with the line. Eq is the magnitude of the injected reactive voltage and Il is the line current. Fig. 2 shows the functional block diagram of the series converter control. The control signal to the supplementary modulation controller (SMC) is the Thevenin angle, DFth which is synthesized from the locally measurable signal i.e., the magnitude of the current through the line in which the series reactive controller is installed. Eqmod is the output of the SMC for the reactive voltage modulation to enhance the damping of low frequency oscillations. Eq follows Eqmod with appropriate time delay. The block diagram of the supplementary modulation controller for SSSC [14] considered in the work presented in this paper is shown in Fig. 3. The controller gains Kq and Xserth are both tunable. Tc and Tp are the controller and plant time constants. The controller is installed on the SSSC to enhance the damping of critical modes. The transfer function of the SSSC with its supplementary modulation controller is:
(1)
where Il is the magnitude of the current in the line where SSSC is connected and Eq is the magnitude of the reactive voltage injected by the SSSC into the system.
<Insert Figure 1>
<Insert Figure 2>
<Insert Figure 3>
Single Machine Equivalent of the Reduced Model Based on Multimodal Decomposition
In multimachine power systems, the SM that mutually interacts with EM near the point of strong resonance is separated by using the multi-modal decomposition concept. Reduced order model of the system is derived based on the concept of multi-modal decomposition [11]. A multi machine power system is considered with the synchronous machines represented by two-axis model (1.1). In this system, the prime-mover dynamics are neglected. For a detailed description of equations of the machine model refer to [3]. The state-space representation for the linearized model of the system can be expressed as:
(2)
(3)
where X is a vector of n state variables, and u = ΔEq and y = ΔIl are the input variable and output variables of the system respectively. Equations (2) and (3) with subscript f for the matrices represents the full system. The swing mode that interacts with the exciter mode near the point of strong resonance is isolated, by applying the concept of multi-modal decomposition [11]. The reduced system of sixth order is obtained by retaining only the modes of interest and the SSSC supplementary modulation controller that contribute to strong resonance can be represented by the state equations
(4)
(5)
The modes, which are not pertinent to the strong resonance phenomenon, are neglected. The block diagram of the reduced system shown in Fig. 4 is equivalent to a single machine infinite bus (SMIB) system. For simplicity, the subscripts 'i' and 'j' are dropped in the figure.
<Insert Figure 4>
From the Heffron-Phillips constants, K1 to K6 and also the gains
and KFr can be expressed in terms of Br matrix can be obtained as in [1]. The system shown in Fig. 4 can be represented by the following equation:
(4)
; matrices M, D, A, B and Bd are defined as in [1].
Strong resonance in presence of SSSC with its supplementary damping controller is investigated for three-machine system, four machine two-area system and four machine practical systems. The generators of all the systems are represented by the two-axis model; for the test system static exciter of KA = 200 and TA = 0.05 is considered while for the practical system KA = 100 and TA = 0.025. The loads are represented as constant impedances.
Analysis of Strong resonance in multi-machine systems
Three-Machine System
A three-machine system shown in Fig. 5 with constant impedance loads is considered to evaluate the performance of the modes interacting near the neighborhood of strong resonance. The system considered is that as in [10] except that a bus (10) is created at the midpoint of the line 7-5. A fixed shunt capacitor of susceptance 0.5 is connected at bus 5 to provide voltage support. For detailed system equations refer to [3].
<Insert Figure 5>
From the eigenvalue analysis the two swing modes obtained for the system without SSSC are shown in Table I:
<Insert Table 1>
It can be observed from Table I that the damping of swing modes is poor. A SSSC with its supplementary modulation controller is connected in the line 5-4 to improve the damping of the modes and the operating value of the output of the SSSC is assumed to be zero. While tuning the SSSC, it is interesting to note that there is an interaction between an exciter mode and a swing mode of frequency 8.58 rad/sec as shown in Fig. 6.
It can be clearly seen that when Xserth is increasing with a fixed change of Kq, the swing mode and exciter mode move towards each other, changing more in damping and a smaller change in frequency, before the point of strong resonance. At the point of strong resonance, they merge and then just after the point, there is drastic change in the direction of eigenvalues. Now, both the modes experience a faster change in frequency, and in damping. Xserth is varying from 0.15 to 0.243 with a magnitude of perturbation of 0.001. When ΔKq is positive exciter mode is going towards right, while swing mode is towards left. When ΔKq is negative, the movement of exciter mode is towards left, while that of swing mode is towards right.
<Insert Figure 6>
<Insert Figure 7>
Fig. 7 shows the presence of strong resonance for the reduced system when Xserth is increasing. Here, before and after the neighborhood of strong resonance, the modes change both in damping and frequency. The strong resonance is observed for the system at the point (0.2502, 11.1257), when Kq = 0.63 and Xserth = 0.1857. On a closer investigation of Figures 6 and 7, it can be found that there is a similarity in the behavior of the eigenvalues for the complete and reduced systems. The movement of exciter mode and the swing mode coincide before and after the point of strong resonance.
Four Machine Two Area System
A four-machine ten-bus system shown in Fig. 8 is considered to evaluate the performance of inter-area mode (IAM) and exciter mode near the neighborhood of strong resonance. The system considered is that as in [10]. In a system, the number of EMs is identical to that of generators. Hence, each EM can be related to one of the generators in the system. The exciter mode interacting with the inter-area mode is identified by computing the participation factors of various state variables in the mode of interest. The absolute values of participation factors are computed for different values of the SSSC damping controller gain and tabulated in Table II. They are computed to find out whether there is any relative change in the contribution of various variables to a particular mode with the change in gain.
While tuning the SSSC installed in one of the AC tie lines between buses 3 and 13, it is observed that there is interaction between an exciter mode and an inter-area mode of frequency 4.4445 rad/sec. From Table II, the participation of variables Eq' and Efd of generator connected to bus 12 in the exciter mode is greater than any other state variables. Hence, this exciter mode can be associated with generator connected to bus 12.
The complete system shown in Fig. 9 when Kq is increasing with ΔXserth = 0, illustrates that EM and SM move towards each other, coalesce and then diverge from each other. The strong resonance is observed for the complete system at the point (-1.499, 3.71) when Kq = 9.60 and Xserth = 0.0214. For this case, Kq is varied from 3 to 20 with a magnitude of perturbations of 0.1 and ΔXserth = 0. Similarly, Fig. 10 shows the asymptotes for the reduced system, and the presence of strong resonance is located at (-1.119, 6.651) when Kq = 7.779 and Xserth = 0.0278. The gain Kq is varied for this system from 7.0 to 9.5 with magnitude of perturbations of 0.01.
<Insert Figure 8>
<Insert Table 2>
<Insert Figure 9>
<Insert Figure 10>
<Insert Figure 11>
<Insert Figure 12>
It can be analyzed from both the plots that, before strong resonance, the damping of EM decreases faster than increase in the damping of SM. After strong resonance, the two modes experience a change in frequency with only a slight change in damping. The modes diverge away from each other approximately at 90o and move apart in opposite directions. Thus, a similar kind of behavior can be noticed from complete and reduced systems.
A comparison is made between the root loci of the complete and reduced systems, shown in Figs. 11 and 12 when Xserth is varying and ΔKq > 0. In both the figures, the behavior of EM and SM coincides before the neighborhood of strong resonance. The destabilization of EM after the strong resonance point is shown more clearly for the complete system than for the reduced system. This difference could be accounted for, due to the approximations involved in the system reduction. For the complete system Xserth is varied from 0.019 to 0.023 with perturbations of 0.0001, while for reduced system Xserth is varied from 0.026 to 0.028 with perturbations of 0.00004.
4 Machine Practical System
The investigation of strong resonance has also been extended to a 4-machine 11-bus Indian practical system, a part of southern grid, and is shown in Fig. 13. Generator connected at Neyveli-4 delivers power to loads at Salem-9 and Sriperumdur-10. A SSSC with its supplementary modulation controller is placed between buses 7 and 10. While tuning the SSSC, it was observed that the swing mode of frequency 7.0283 rad/sec interacts with an exciter mode indicating the presence of strong resonance. By the method of participation factors not shown, it is evaluated that this exciter mode is associated with generator connected to bus 4.
<Insert Figure 13>
A comparison is made between asymptotes for the reduced system and the root loci of full system shown in Figs. 14 and 15. In both figures, Kq is increasing for a fixed value of Xserth. In the full system, Kq is varied from 0.7 to 1.2 while in reduced system Kq is varied from 0.8 to 1.2. The magnitude of perturbations effected in Kq is 0.01 for full system while it is 0.001 for reduced system. Similarly, Figs. 16 and 17 show the asymptotes for the reduced system and the root loci of full system, in which Xserth is increasing for a fixed value of Kq. In full system (Fig. 17), the variation of Xserth is from 0.025 to 0.035 while in reduced system (Fig. 16), Xserth is varied from 0.032 to 0.035 for both ΔKq > 0 and ΔKq < 0. The magnitude of perturbations effected in Kq is 0.0001 for full system while it is 0.00001 for reduced system.
<Insert Figure 14>
<Insert Figure 15>
<Insert Figure 16>
The strong resonance for the full system is observed at the point (-2.61 ± j 6.825) when Kq = 0.8615 and Xserth = 0.0336 while for the reduced system it is found at (-2.55 ± j 7.5565) when Kq = 1.010 and Xserth = 0.0337. This also shows that there is not much of difference in the controller parameters and point of strong resonance between full and reduced systems.
This behavior of strong resonance is identical for full and reduced order systems. It shows that one of the modes become unstable after the point of strong resonance.
<Insert Figure 17>
Discussion
3-machine and 4 machine 2 area systems
As Kq is varied for ΔXserth > 0 and ΔXserth < 0, before the point of strong resonance the two modes approach one another but after the point of strong resonance:
1) For ΔXserth > 0, SM is moving towards left and is stabilized while EM is moving towards right and is destabilized.
2) For ΔXserth < 0, EM is moving towards left and is stabilized while SM is moving towards right and is destabilized.
As Xserth is varied for ΔKq > 0 and ΔKq < 0, before the point of strong resonance the two modes approach one another while after the point of strong resonance:
1) For ΔKq > 0, EM is moving towards right and is getting destabilized while SM is moving towards left and is being stabilized.
2) For ΔKq < 0, SM is moving towards right and is getting destabilized while EM is moving towards left and is being stabilized.
In the neighborhood of strong resonance the two systems are complementary to each other and their qualitative behavior is almost identical for full and reduced order systems. While, in a 3-machine system either one of the modes is destabilized, in the case of a 4-machine system both modes are destabilized for the reduced system after the point of strong resonance.
4 machine 11 bus practical system
As Kq is varied for ΔXserth > 0 and ΔXserth < 0, before the point of strong resonance the two modes approach one another but after the point of strong resonance:
1) For ΔXserth > 0, EM is moving towards left and is stabilized while SM is moving towards right and is destabilized.
2) For ΔXserth < 0, SM is moving towards left and is stabilized while EM is moving towards right and is destabilized.
As Xserth is varied for ΔKq > 0 and ΔKq < 0 before the point of strong resonance the two modes approach one another while after the point of strong resonance:
1) For ΔKq > 0, SM is moving towards left and is stabilized while EM is moving towards right and is destabilized.
2) For ΔKq < 0, EM is moving towards left and is stabilized while SM is moving towards right and is destabilized.
For all the three systems considered, the nature of SM or IAM and EM around the point of strong resonance is similar. It is so under the conditions of 1) Kq varying with ΔXserth = 0, and 2) Xserth varying with ΔKq = 0. However in these observations, there is a shift in the angular orientation and shape of the root loci, rate of change of damping or frequency and the controller parameters between complete and reduced order systems. These differences could be attributed to the trivial approximations involved while reducing the system size.
Conclusion
The phenomenon of strong resonance is investigated by applying multi-modal decomposition. This work evaluates the modes liable for strong resonance in presence of the SSSC with its SMC. Case studies have been carried out on 3-machine, 4-machine test systems as well as a 4-machine Indian practical system.
The interaction of swing mode and the exciter mode has been identified using the concept of participation factors. The geometric perturbations of the two modes in the vicinity of strong resonance is investigated for reduced systems and compared with the root loci of the complete systems with SSSC. The results show that characteristic performance of the coupled modes is in conformity in both situations.
The above research strongly suggests that the effect of strong resonance on these interacting modes should be considered while designing the damping controller for the SSSC.
The appropriate choice of control parameters for the optimal tuning of the controller, accounting the effect of modal interaction is vital for its design; in order to ensure a high performance and reliable power system.
appendix a
MATRICES OF VECTOR DIFFERENTIAL EQUATION(10)
The reduced system represented by (8) and (9) with SSSC and its damping controller is expressed by a vector differential equation (10) in section V. The output (9) of the system with the SSSC damping controller can also be written as
(11)
where C = [Cr1 Cr3]. Cr2 = Cr4 = 0 and Cri is the ith element of Cr. Equation (1) gives the mathematical model of the SSSC damping controller. From this equation, we have
(12)
Substituting (12) in (11), it is easy to obtain an equation relating ΔEq and q as
(13)
where
substituting of (13) in (10) gives
(14)
where
The phenomenon of strong resonance with SSSC for the reduced system is studied. The relevant hyperbolae are constructed by making suitable modifications based on the theory suggested by Seiranyan [5] and the work presented by Padiyar [2] and then the asymptotic movement of the two oscillatory modes is then studied as one of the parameters varies. In the present paper, the system and controller dynamics are represented by vector-matrix differential equation as given in (14). The elements of the matrices A1, A2, A3, A4 and A5 are functions of the components of the parameter vector p = (Kq, Xserth).
