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Feasibility analysis on FESS damping for power system oscillation.

I. INTRODUCTION

In the recent years, researchers all over the world have done a lot of work on using FESS to improve the stability of power system consisting of doubly-fed induction motors (DFIM) [1], [2]. However, there are few research papers on applying FESS to damp multi-mode oscillation in multi-machine system. The usual solution is to configure stabilizers in multi-machine system according to each oscillation mode respectively [3]. However, there is a problem called "eigenvalue deviation" because the stabilizers would affect each other [4]. To address it, some researchers put forward optimization algorithm to tune the parameters of the stabilizers [5], [6].

FESS has two independent loops to control active power and reactive power respectively. Because both of its loops can be configured with stabilizers, FESS device has the ability to damp multi-mode oscillation in power systems.

This paper mainly describes the theory on damping power system multi-mode oscillation using FESS. Firstly, damping torque analysis is applied to prove that FESS has the ability to damp the multi-mode oscillation. Then, control loops and feedback signals are chosen based on control theory, and particle swarm optimization (PSO) is used to tune the parameters of the stabilizers. Finally, a four-machine power system case was investigated to testify the validity of the proposed method.

II. LINEARIZED MODEL OF THE MULTI-MACHINE SYSTEM WITH FESS

Compared with other energy storage technology, FESS, composed by doubly fed induction motor (DFIM), is paid more attentions owing to its advantage in economy and practice [2], [7], [8]. As energy is stored in its rotating rotor, FESS can exchange energy with the system via DFIM by adjusting the speed of the flywheel (Fig. 1).

FESS can be described as a third-order dynamic model. Generally, Stator flux orientation is chosen as its excitation control strategy because it can accomplish the decoupling control of active and reactive power. Meanwhile the reactive power can be replaced by voltage control.

When FESS is used to damp low frequency oscillation, stabilizers can be configured for its active power and voltage control loops. The schematic diagram is shown in Fig. 2. where: Ks is the gain of stabilizer; KPX and KIX are respectively proportional integral factors. It is assumed that FESS damping controllers are configured in active power and voltage control loops respectively and the output signals are Vsp and Vsu.

Combine the network algebraic equations and with linearized model of FESS, the linear equation of whole system can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [DELTA][delta] is the deviation in generator power angle; [DELTA][omega] is the deviation in generator speed; [DELTA]Z is the state variables of generator (except power angle and speed), including state variables of FESS devices (except stabilizer). [B.sub.JP] and [B.sub.Ju] are respectively the transfer functions from the input signals ([[DELTA]V.sub.sp] and [[DELTA]V.sub.su]), to the state variables.

(1) can also be demonstrated as Fig. 3.

The output signal that is the feedback signal can be expressed in the following equation

[DELTA]y = C[[[DELTA][delta] [DELTA][omega] [DELTA]z].sup.T], (2)

where C is the transfer function from the state variables to y.

[G.sub.p](s) and [G.sub.u](s) are respectively the transfer functions in active power and voltage control loops attached with damping controllers, then [[DELTA]V.sub.sp] and [[DELTA]V.sub.su] can be written as (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [[DELTA]y.sub.p] and [[DELTA]y.sub.u,] are respectively the deviations in active power and voltage damping controllers' input signals, which can be achieved from (2).

(1)-(3) are the linearized model of the multi-machine system with FESS.

III. FEASIBILITY ANALYSIS ABOUT MULTI-MODE OSCILLATION DAMPING WITH FESS

FESS has two independent loops, controlling active power and reactive power respectively. Both loops can be configured with damping controllers independently.

According to (1) and Fig. 3, total influence to the mode [[lambda].sub.i] of two additional stabilizers can be calculated as follows [9]

[DELTA][[lambda].sub.i] = [N.summation over (j=1)] [S.sub.ij] ([H.sub.pj][angle][[phi].sub.pj][[DELTA]G.sub.p]([[lambda].sub.i])). (4)

where: [S.sub.ij] is the sensitivity coefficient, defined as mode [lambda]i partial derivative of [T.sub.Dij] , which is the damping torque of electro-mechanical oscillations in the j-th generator. [H.sub.pj] [angle] [[phi].sub.pj] and [H.sub.uj] [angle] [[phi].sub.uj] are the forward channel from the damping controllers to electro-mechanical oscillations in the j-th generator.

From (4), it is obvious that the damping controllers supply damping torque to N generators with through N channels firstly and then N generators distribute the damping torque to each mode in the system via sensitivity coefficient. It shows how these two damping controllers restrain the multi-mode oscillation, and prove the feasibility that FESS can suppress the multi-mode oscillation.

IV. TUNING PARAMETERS OF FESS DAMPING CONTROLLERS

For damping multi-mode oscillation in the power systems, it is necessary to select appropriate FESS control loops, feedback signals, and parameters tuning of the damping controllers based on control-ability and observe-ability. Furthermore, owing to its high efficiency and the ability to find the optimal global solution, PSO algorithm is adopted to tune the parameters [10].

According to the control theory, the performance index [b.sub.iK] of mode [[lambda].sub.i] can be written as (5)

[b.sub.iK] = [absolute value of [W.sup.i.sub.T] [B.sub.k]], (5)

where [W.sub.i.sub.T] is the left eigenvector of system state matrix A corresponding to mode i. [B.sub.K] is the column vector in (1) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, controlling [DELTA][V.sub.sp] and [DELTA][V.sub.su], respectively.

The observe-ability index [c.sub.iK] of mode [[lambda].sub.i] can be written as follows

[c.sub.iK] = [absolute value of [CV.sub.i]], (6)

where [V.sub.i] is the right eigenvector of system state matrix A corresponding to mode i.

From (5) and (6), control loops and feedback signals can be selected according to the performance indicators of controllability and observe-ability, respectively.

From (4), it is obvious that two damping controllers influence each other when they are used to restrain multi-mode oscillation. Therefore, PSO algorithm is adopted to tune the parameter of the damping controllers in FESS.

The structure of the controller is shown in Fig. 2. The time constants of measuring sector and DC block sector, marked as T5 and T6, are given. Lead and lag sectors have the same time constants, that is to say, [T.sub.1] = [T.sub.3], [T.sub.2] = [T.sub.4]. Hench each controller has three unknown parameters, time constant [T.sub.1], [T.sub.2] and gain Ks. Then there are totally six unknown parameters. The objective function is maximizing the minimum of damping ratios of two weak damping modes, marked as [[xi].sub.1], [[xi].sub.2]. So it can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [K.sub.[alpha]] is the gain of two controllers; [T.sub.[alpha]] is the time constant of the lead and lag links in the controllers.

(7) is a typical optimization problem with constraints, which can be dealt by PSO to achieve coordinated tuning.

V. CASE STUDY

A four-machines power system was investigated in this paper [11], shown in Fig. 4, assuming FESS connected at node 7. According to the eigenvalue, there are three weak damping modes:

1) Regional mode, [[lambda].sub.1] = 0.05259 [+ or -] j3.9484, the damping ratio is -0.013 (unstable).

2) Part mode between generator G1 and G2, [[lambda].sub.2] = -0.3167 [+ or -] j6.0250, the damping ratio is 0.052 (weak damping mode).

3) Part mode between generator G3 and G4, [[lambda].sub.3] = -0.3070 [+ or -] j6.1364, the damping ratio is 0.050 (weak damping mode).

Since FESS is set in node 7, which is far from generator G3 and G4, the effect towards mode 3 will be not so well. Therefore, FESS is used to restrain [[lambda].sub.1] and [[lambda].sub.2].

According to (5), the controllability indices of FESS active and reactive power control loops towards [[lambda].sub.1] and [[lambda].sub.2] can be calculated and summarized in Table I.

The results in Table I show that:

1) FESS can control regional mode [[lambda].sub.1] effectively, while it is not very ideal to control part mode [[lambda].sub.2] .

2) Both indices show that active power control loop performs better than voltage control loop.

According to (6), the observe-ability indices of FESS to regional mode [[lambda].sub.1] and part mode [[lambda].sub.2] can be calculated and shown in Table II.

From Table II, oscillation power [P.sub.78] of line 7-8 is suitable to be taken as the feedback signal to restrain regional mode [[lambda].sub.1], and the integral of the power difference between line 5-7 and line 2-7 ([integral](P57-P27)) is suitable to be taken as the feedback signal to restrain part mode [[lambda].sub.2].

Taking maximum of the minimum damping ratios of [[lambda].sub.1] and [[lambda].sub.2] as the objective function, stabilizer's parameters can be achieved based on PSO algorithm, which is shown in Table III.

Applying such parameters into operation control, the eigenvalues of [[lambda].sub.1] and [[lambda].sub.2] can be recalculated as Table IV shows. It is obvious that oscillation of [[lambda].sub.1] and [[lambda].sub.2] are restrained.

Considering there was a three-phase short-circuit fault occurred at bus 8 which lasts 0.1s, simulation results shown in Fig. 5 also verify the validity of the proposed stabilizers.

VI. CONCLUSIONS

It has been proved in this paper that FESS can restrain multi-mode oscillation in power system based on the analysis of damping torque.

Optimization algorithm can be utilized for parameters tuning for the design of FESS stabilizer.

http://dx.doi.org/ 10.5755/j01.eee.19.2.1699

REFERENCES

[1] J. McGroarty, J. Schmeller, R. Hockney, M. Polimeno, "Flywheel energy storage system for electric start and an all-electric ship", in Proc. of IEEE International Conference on Electric Ship Technologies Symposium, Philadelphia, 2005, pp. 400-406.

[2] L.-J. Shi, L. Zhang, M.-H. Zhuang, G.-Q. Tang, "Applications of FESS in power system muti-mode oscillations", in Proc. of the 4th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), Weihai, 2011, pp. 1732-1736.

[3] K. Rahmani, M. S. Naderi, G. B. Gharehpetian, "Damping of parallel AC-DC power system oscillations using LQG/LTR controller approach", in Proc. of 19th Iranian Conference on Electrical Engineering (ICEE), Iran, 2011, pp. 1-6.

[4] H. B. Gooi, E. F. Hill, M. A. Mobarak, D. H. Thorne, T. H. Lee, "Coordinated Multi-Machine Stabilizer Settings Without Eigenvalue Drift", IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 8, pp. 3879-3887, 1981. [Online]. Available: http://dx.doi.org/10.1109/TPAS.1981.316983

[5] M. A. Abido, "Optimal design of power-system stabilizers using particle swarm optimization", IEEE Transactions on Energy Conversion, vol. 17, no. 3, pp. 406-413, Sep. 2002. [Online]. Available: http://dx.doi.org/10.1109/TEC.2002.801992

[6] M. O. Hassan, S. J. Cheng, Z. A. Zakaria, "Design and parameters optimization of power system stabilizer to improve power system oscillations" in Proc. of the 2nd International Conference on Computer and Automation Engineering (ICCAE), Singapore, 2010, pp. 107-111.

[7] P. Balciunas, P. Norkevicius, "Nature Electrical Energy Conversion Processes Experimental Research of Wind Micro Power Plant", Elektronika ir Elektrotechnika (Electronics and Electrical Engineering), no. 9, pp. 57-60, 2010.

[8] V. Adomavicius, V. Kepalas, "Control of Wind Turbine's Load in order to maximaize the Energy Output", Elektronika ir Elektrotechnika (Electronics and Electrical Engineering), no. 8, pp. 71-76, 2008.

[9] W. K. Marshall, W. J. Smolinski, "Dynamic Stability Determination by Synchronizing and Damping Torque Analysis", IEEE Transactions on Power Apparatus and Systems, vol. PAS-92, no. 4, pp. 1239-1246, 1973. [Online]. Available: http://dx.doi.org/10.1109/TPAS. 1973.293806

[10] J. L. Fernandez-Martinez, E. Garcia-Gonzalo, "Stochastic Stability Analysis of the Linear Continuous and Discrete PSO Models",

IEEE Transactions on Evolutionary Computation, vol. 15, no. 3, pp. 405-423, 2011. [Online]. Available: http://dx.doi.org/10.1109/TEVC.2010.2053935

[11] Q. Xu, H. Zang, L. Shi, "Stabilizer design based on flywheel energy storage system with multiple operations for multi-machine power systems", Journal of Renewable and Sustainable Energy, vol. 4, pp. 1-10, 2012. [Online]. Available: http://dx.doi.org/10.1063/1.3690956

Qingshan Xu (1), Linjun Shi (2), Longhuan Shen (3)

(1) Engineering Research Center of Motion Control, Ministry of Education, Southeast University, Sipailou 2#, Nanjing, China 210096

(2) School of Energy and Electrical Engineering, Hohai University, Xikang Road 1#, Nanjing, China 210098

(3) School of Electrical Engineering, Xi'an Jiaotong University, Xianning West Road 28#, Xi'an, China 710049 xuqingshan@seu.edu.cn

Manuscript received May 14, 2012; accepted October 2, 2012.

The research is financially supported by National High Technology Research and Development Program of China (863 Program) (No.2012AA050214), Natural Science Foundation of Jiangsu Province (No. BK2012753) and the Fundamental Research Funds for the Central Universities


TABLE I. CONTROLLABILITY INDICES OF DIFFERENT CONTROL LOOP.

FESS control     Controllability index of   Controllability index
loop                 [[lambda].sub.1]        of [[lamda].sub.2]

Active power             0.24644                   0.05977
control loop

Reactive power           0.21484                   0.01561
control loop

TABLE II. OBSERVABILITY INDICES OF DIFFERENT FEEDBACK SIGNALS.

Mode               P78 as    [integral]([P.sub.57]-[P.sub.27])
                  Feedback              as Feedback
                   signal                 signal

Regional mode      0.7631                 0.1364
[[lambda].sub.1]

Part mode         0.02659                 6.99670
[[lambda].sub.2]

TABLE III. PARAMETERS OPTIMIZATION OF FESS STABILIZERS.

Parameter of   Control loop of    Control loop of
stabilizers    [[lambda].sub.1]   [[lambda].sub.2]

T1 = T3             0.023              0.312
T2 = T4             0.054              0.105
Ks                   5.11               1.23

TABLE IV. INFLUENCE OF DAMPING BY FESS STABILIZERS.

             [[lambda].sub.2]

Eigenvalue                  Damping ratio

-0.3167 [+ or -] j6.0250        0.052
-0.5986 [+ or -] j5.8140        0.102

             [[lambda].sub.1]

Eigenvalue                  Damping ratio

-0.05259 [+ or -] j3.9484      -0.013
 -0.3513 [+ or -] j3.3098       0.106
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Author:Xu, Qingshan; Shi, Linjun; Shen, Longhuan
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:9CHIN
Date:Feb 1, 2013
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