Small Signal Stability Analysis of Multi-Machine Power Systems Interfaced with Micro Grid

Small Signal Stability Analysis of Multi-Machine Power Systems Interfaced with Micro Grid

1. Padma Subramanian

   EEE Department

   Professor and Dean, Faculty of Electrical Engineering Director- Centre for Robotics and Automation AMET University, Kanathur, Chennai, India

   Abstract This paper work presents a study of small signal stability analysis of multi-machine power systems interfaced with micro grid. Modeling of DFIG for Wind Energy Conversion System (WECS), marine current energy system and, and PV module is presented. A procedure for incorporating Wind Energy Conversion System, Marine Current Energy System and PV system into multi-machine power systems is presented. A program is developed in MATLAB environment and effectiveness of developed program is tested in a standard IEEE-9 bus system. The small signal stability of the multi- machine power systems interfaced with micro grid is analyzed and the results are presented.

   Keywords Wind Energy Conversion System, Marine Current Energy Conversion System, PV System, Multi- Machine Power Systems, Eigen-Value Analysis, Small Signal Stability Analysis

   1. INTRODUCTION

      Renewable energy is one of the sizzling themes in the entire world today due to the fast and huge consumption of fossil fuels. The ocean covers more than 70% surface of the earth, the wind energy above the sea surface and oceanic energy under the seawater can be captured simultaneously to generate large electric power[2,4,11,12]. The wind energy and oceanic energy can be effectively integrated together to deliver electric power to the loads. The utilization of marine current turbines offers an exciting proposition for the extraction of energy from marine currents[1]. Wind, tidal and PV system has a higher reliability for maintaining a continuous power than any other individual sources[2,14]. Compared to external grid, micro-grid is a single controllable unit, that link multiple distributed power generation sources into a small network The modelling of PV cell in simplified equivalent circuit with output elements and the stability of PV system using Eigen-value analysis was discussed by Hun-Chul Seo . Shan ying li presented the analysis of small signal stability of grid-connected doubly fed induction generators .The detailed model of grid-connected DFIG wind turbine is firstly established, and the Eigen-values are classified and characterized based on participation factors.

      K. Harinee EEE Department

      Jerusalem College of Engineering Chennai, India

      This paper presents a modeling of power system and micro grid for small signal stability analysis, step by step procedure for small signal stability analysis of micro grid interfaced with multi-machine system and state space model for multi-machine system with micro grid. The small signal stability analysis of multi-machine system with micro grid is presented using Eigen-value analysis.

      This paper comprises of the following sections: Section 2 deals with the modeling of power system. In section 3, the small signal stability analysis of multi-machine system with micro grid is presented. In section 4, the results and discussion of the test system are presented. Conclusion is presented in section 5.

   2. POWER SYSTEM MODELING

      In this section, modeling of synchronous generator in power system and micro grid components for small signal stability analysis are presented.

      1. Synchronous Generator Model

         In multi-machine model, a synchronous machine or group of synchronous machines connected to a larger system through one or more power lines as shown in Fig-1.

         Fig-1: General m machine n bus system

         Synchronous Generators in multi-machine power systems are modeled as classical machine model and variable voltage behind transient reactance model.

         Neglecting saliency, the stator of a synchronous machine is represented by the equivalent circuit shown in Fig-2. The block diagram of excitation system is shown in Fig-3.

         (3)

         (4)

         (5)

         (6)

         Where and refers rotor angle and speed, is quadrature voltage behind transient reactance, is terminal voltage transducer, H is inertia constant, KD is

         damping coefficient.

         Where,

         (7)

         (8)

         Fig-2: Stator equivalent circuit

         Fig-3: Simplified block diagram of excitation system

         The linearized equations of classical machine model for the small signal stability analysis in state variable form are represented by the following equations[6,7,8]

         (1)

         (2)

         The linearized equations of variable voltage behind transient reactance model for the small signal stability analysis in state variable form are represented by the following equations[6,7,8]

         (9)

         (10)

         (11)

         (12)

         Where refers direct axis synchronous is direct axis transient reactance, is direct axis open circuit transient time constant and refers quadrature

         axis synchronous reactance, B is susceptance and G is conductance.

      2. Doubly Fed Induction Generator Model for Wind and Marine Farm

         DFIG is an induction-type generator. The d-axis and q- axis equivalent circuit of doubly fed induction generator for wind and marine farm is shown in Fig-4 (a) and (b).

         (17)

         (18)

         (19)

         (20)

         Fig-4: (a) d-axis equivalent circuit of doubly fed Induction generator

         (b) q-axis equivalent circuit of doubly fed induction generator

         1. Electrical Equations

            The linearized electrical differential equations of DFIG in wind farm for small signal stability analysis in state variable form are represented by the following equations

            (13)

            (14)

            Where, – electrical base speed, – synchronous speed, – q-axis and d-axis voltage of stator,

            – q-axis and d-axis current of stator for

            marine farm, – q-axis and d-axis current of rotor for marine farm, – stator and rotor inductances for marine farm, – mutual inductances for

            marine farm, – rotor and stator resistances for

            marine farm.

         2. Mechanical Equations

         A two-mass drive train model is used to get a more accurate response from the wind turbine and marine turbine. Two mass drive train model is shown in Fig-5.

         (15)

         (16)

         Where, – electrical base speed, – synchronous speed, – q-axis and d-axis voltage of stator,

         – q-axis and d-axis current of stator for

         wind farm, – q-axis and d-axis current of rotor for wind farm, – stator and rotor inductances for wind farm, – mutual inductances for wind farm, – rotor and stator resistances for wind

         farm.

         The linearized electrical differential equations of DFIG in marine farm for small signal stability analysis in state variable form are represented by the following equations

         Fig- 5: Two mass drive train model

         The linearized mechanical differential equations of DFIG in wind farm for small signal stability analysis in state variable form are represented by the following equations

         (21)

         (22)

         (23)

         Where, turbine and rotor speeds for wind farm, – shaft torsional angle for wind farm,

         -drive train damping coefficient and

         shaft stiffness for wind farm, turbine

         and generator inertia for wind farm, -mechanical torque for wind farm, Tew – electrical torque for wind farm.

         The linearized mechanical differential equations of DFIG in marine farm for small signal stability analysis in state variable form are represented by the following equations

         (24)

         (25)

         (26)

         Where, turbine and rotor speeds for marine farm, – shaft torsional angle for marne farm, -drive train damping coefficient and shaft stiffness for marine farm,

         turbine and generator inertia for marine farm, – mechanical torque for marine farm, Tem electrical torque for marine farm.

      3. PV Cell Model

         PV arrays are built up with combined series/parallel combinations of PV solar cells, which are usually represented by a simplified equivalent circuit model such as the one given in Fig-6.

         Fig-6: Simplified equivalent circuit of PV cell

         The linearized differential equations of PV cell for small signal stability analysis in state variable form are represented by the following equations

         (27)

         (28)

         Where, Rs- array resistance, Iph , I0 , are constants, IC and VC are current and voltage through PV cell.

      4. Load Model

      The load is modeled as constant impedance load. The load (P + jQ) is represented as constant impedance load. The equation of load for small signal stability analysis are represented by equation (29).

      ZL=VL/IL (29)

      Where VL = Load Voltage, IL = Load Current, ZL = Load impedance.

   3. SMALL SIGNAL STABILITY ANALYSIS

      Small signal stability analysis is performed by linearizing the system equations at the operating point and Eigen-value analysis. The system equations are described in the following general form

      X=f(X,Z,U), Z=g(X,U) (30)

      Where X, Z and U are the vectors of state variables, control variables and input variables respectively. After performing the linearization, the following relation is derived.

      (31)

      The state variable X is given as Classical machine model:

      X=

      Variable voltage behind transient reactance model X=

      Where A is the system state matrix. This matrix is then used for calculating the system Eigen-values.

   4. RESULTS AND DISCUSSION

      In this section, results of small signal stability analysis of multi-machine power systems interfaced with micro grid carried out on standard IEEE 9 bus system.

      A. IEEE-9 bus system

      In this paper, standard IEEE-9 bus system is considered for small signal stability analysis. Data for the 9 bus system is provided in appendix. The single line diagram of micro grid interfaced standard IEEE- 9 bus system is shown in Fig- 7.

      The load is represented by constant impedance load. The losses in transmission lines are neglected .The mechanical power is assumed to be constant.

      The small signal stability analysis is carried out by interfacing micro grid including WT, MCT and PV module between 7 and 8 bus of IEEE-9 bus system. The small

      signal stability analysis result with and without micro grid obtained for classical machine model and variable voltage behind transient reactance model in synchronous generator is tabulated in Table-1 and Table-2.

      Fig-8: Single line diagram of micro grid interfaced in IEEE-9bus system

      Table -1: Eigen-values for IEEE-9 bus system without micro grid

      No of state variables	Eigen-values for classical machine model	Damping ratio	Eigen-values for variable voltage behind transient reactance model	Damping ratio
      1	0		0
      2	-0.5236		-0.5236
      3	0		0
      4	-1.5015		-1.5015
      5	0		0
      6	-2.1276		-2.1276
      7			-8.3491 + 3.1436i	0.9358
      8			-8.3491 – 3.1436i	0.9358
      9			-2.5634
      10			-14.2615
      11			-8.4882 + 3.8651i	0.9358
      12			-8.4882 – 3.8651i	0.9358

      Table-2: Eigen-values for IEEE-9 bus system with micro grid

      No of state variables	Eigen-values for classical machine model *10^4	Damping ratio	Eigen-values for variable voltage behind transient reactance model*10^4	Damping ratio
      1	-0.0000 + 0.0007i		0
      2	-0.0000 – 0.0007i		-0.0001
      3	-0.0003		0
      4	-0.0044 + 1.0329i	0.0042	-0.0002
      5	-0.0044 – 1.0329i	0.0042	0
      6	-0.0012 + 1.0781i	0.0011	-0.0002
      7	-0.0012 – 1.0781i	0.0011	-0.0006
      8	-1.5458		-0.0011
      9	-0.0123		-0.0003
      10	-0.0003 + 0.6060i	0.0004	-0.0014
      11	-0.0003 – 0.6060i	0.0004	-0.0006
      12	-0.0011 + 0.5608i	0.0019	-0.0011
      13	-0.0011 – 0.5608i	0.0019	-0.0003 + 0.6060i	0.0004
      14	-0.0000 + 0.0006i		-0.0003 – 0.6060i	0.0004
      15	-0.0000 – 0.0006i		-0.0011 + 0.5608i	0.0019
      16	-0.0174		-0.0011 – 0.5608i	0.0019
      17	0		-0.0000 + 0.0006i
      18	-0.0001		-0.0000 – 0.0006i
      19	0		-0.0174
      20	-0.0002		-0.0012 + 1.0781i	0.0011
      21	0		-0.0012 – 1.0781i	0.0011
      22	-0.0002		-0.0044 + 1.0329i	0.0042
      23			-0.0044 – 1.0329i	0.0042
      24			-0.0000 + 0.0007i
      25			-0.0000 – 0.0007i
      26			-0.0003
      27			-0.0123
      28			-1.5458

      From Table-3 and Table-4, it is observed that the system remains stable after interfacing micro grid into the system. Based on the Eigen-values, it is concluded that the small signal stability of the system is affected after interfacing micro grid into the system.

   5. CONCLUSION

In the project work, the small signal stability analysis of multi-machine power systems interfaced with micro grid is presented. A program is developed using MATLAB .The effectiveness of the developed program is tested using a standard IEEE-9 bus system. The smal signal stability results show that the system remains stable and stability of the system is affected after interfacing micro grid into the system.

APPENDIX

IEEE-9 BUS SYSTEM DATA

Generator Data

Bus Data

Line Data

Stored Energy H1 = 9.55sec

H2 = 3.33sec H3 = 2.35sec

Frequency f = 60Hz

MICRO GRID DATA

Wind Farm Data:

Air density=1.223Kg/ m3; Blade pitch angle =30 degree; Power coefficients C1=0.5, C2=116, C3=0.4, C5=5, C6=21

;P=2MW; Rs=0.00706 p.u.; Xls=0.171 p.u; Xm=2.9p.u;

Rr=0.005p.u; Xlr=0.156 p.u ; Hhw – 3.5s; Hgw – 0.5s; Dhgw- 0.1p.u ; Khgw -0.1p.u.

Marine Current Farm Data:

Water density=1025 Kg/ m3; Blade pitch angle =0 degree; Power coefficientsd1=0.18;d2=85;d3=0.38;d4=0.25;d5=0.5; d6=11;d7=10.9;d8=0.08;d9=0.01;P=2.5MW;Rs=0.01619p. u;Xls=0.1335 p.u; Xm=3.99p.u; Rr=0.12p.u;

Xlr=0.1121p.u; Hhm – 2.5s; Hgm – 0.5s; Dhgm- 0.1p.u ; Khgm -0.1p.u.

PV Generation System Data

e= 1.602*10^-19 C; K = 1.38*10^-23 J/K; P=130 w; Tc=298K; Iph=5.14;I0=0.0002A; Rs=0.001ohm; L=1 ; C=10mH.

REFERENCES

1. Batten W.M.J, Bahaj A.S, Molland A.F, Chaplin J.R. (2006) Hydrodynamics of marine current turbines Renew. Energy Vol. 31 pp. 249256.

2. Elghali S.E.B, Balme R, Saux K.L, Benbouzid M.E.H Charpentier J.F, Hauville F. (2007) A simulation model for the evaluation of the electrical power potential harnessed by a marine current turbine IEEE J. Oceanic Eng Vol. 32 pp. 786 797.

3. Hun-Chul Seo and Chul-Hwan Kim. (2012) Analysis of stability of PV system using eigen-value according to the frequency variation and requirements of Frequency protection Journal of Electrical Engg. and Technology Vol. 7 No.4 pp. 480-485.

4. Li Wang and Chia-Tien Hsiung. (2011) Dynamic stability improvement of an integrated grid-connected offshore wind farm and marine current farm using a STATCOM IEEE Trans. on Power Systems Vol 26 No.2 pp. 690-698.

5. Muyeen S.M, Takahashi R, Murata T, TamuraJ.(2010) A variable speed wind turbine control strategy to meet wind farm grid code requirements IEEE Trans. Power Systems Vol. 25 No.1 pp. 331- 340.

6. Kundur.P.(2001) Power system stability and control Tata McGraw- Hill publishing company limited New Delhi.

7. Ramanujam.R. (2009) Power system dynamics PHI learning private limited New Delhi.

8. Sauer.P.W and Pai.M.A. (2001) Power system dynamics and stability Pearson publishing company limited New Delhi.

9. Shan-Ying Li , Yu Sun ,Tao Wu. (2010) Analysis of small signal stability of grid- connected doubly fed induction generators in proc. of IEEE Int.Conf.on power systems.

10. Teleke S, Baran M.E, Huang A.Q, Bhattacharya S, Anderson L. (2009) Control strategies for battery energy storage for wind farm Dispatching IEEE Trans. Energy Convers. Vol.24 pp. 725732.

11. Wang L, Chen S.-S, Lee W.-J, Chen Z. (2009) Dynamic stability enhancement and power flow control of a hybrid wind and Marine current farm using SMES IEEE Trans. Energy Conversation vol. 24 pp. 626 639.

12. Wang L, Yu J.Y, Chen Y.T. (2011) Dynamic stability improvement of an integrated offshore wind and marine-current farm using a flywheel energy storage system IET

    Renew. Power Generation Vol.5 pp. 387-396.

13. Yang. B, Makarov.Y, Desteese.J, Viswanathan.V, Nyeng.P, McManus.B, Pease.J. (2008) On the use of energy storage technologies for regulation services in electric power systems with significant penetration of wind energy Proc. 5th Int. Conf. European Electricity Market pp. 2830.

14. Yun Tiam Tan, Daniel .S. Kirschen, Nicholas Jenkins. (2004) A Model of PV Generation Suitable for Stability Analysis IEEE Trans. on Energy Conversion Vol.19 No. 4.

Dr. D. Padma Subramanian received B.Tech. in Electrical and Electronics Engineering and M.Tech. in Electrical Power Systems with Honours from Govt. Colleg of Engineering, Thrissur in 1990 and 1992 respectively. She has done full time doctoral research from College of

Engineering,Guindy,Anna university during 2004-2007. She has a total of over 21 years of experience in the field of teaching industry and research. Dr. Padma Subramaian has more than 30 publications spanning around various international/ national journals and proceedings with good impact factor. She is serving as a member of editorial board for a number of international journals. She is an active researcher in the field of power systems non linear dynamics, FACTS applications to power systems, modeling and analysis of micro grid, grid integration issues of renewable energy sources and self healing networks for smart grid. She has completed couple of funded projects as Principal Investigator. She is a member of doctoral committee for research in various universities and approved supervisor for guiding Ph.D/M.S in many universities including Anna University. Presently she is working as Dean-Faculty of Electrical Engineering and Director- Centre for Robotics and Automation at AMET University, Kanathur, Chennai.

Harinee.K, born in Tamil Nadu, completed BE degree in electrical and electronics engineering from Sri Sairam Engineering College Chennai, India.Completed ME in Power Systems in Valliammai Engineering College, Chennai, India . Currently working as assistant professor at Jerusalem College of Engineering.