Small Signal Stability Enhancement of Distribution System with Distributed Generation using Exact Model Matching

Small Signal Stability Enhancement of Distribution System with Distributed Generation using Exact Model Matching

P. Vidya sagar

PG scholar

Electrical Engineering Department

National Institute of Technology, Calicut, Kerala, India

Sunil Kumar T. K.

Assistant Professor Electrical Engineering Department

National Institute of Technology, Calicut, Kerala, India

Abstract the market deregulation of power sector have encouraged the distributed generation, mainly in distribution systems. Due to the dynamics of distributed generator and dynamics of large loads the distribution systems are facing different types of stability issues. This paper examines the small signal stability of a distributed generation based distribution system. The different oscillatory modes in the system are investigated. A control methodology using exact model matching technique to support small signal stability of a distribution system is proposed. The effectiveness of controllers is illustrated by obtaining closed loop response of the system for input disturbance and comparing the closed loop response with response of desired model.

Keywords distributed generation , exact model matching, participation factors, small signal stability,

I. INTRODUCTION

Recent trends of legislative changes around the globe are likely to increase the penetration of distributed generation (DG) units into power systems. The integration and high penetration of DG units into a distribution system could introduce a number of key issues including oscillatory stability which is also referred to as small signal stability [1].oscillatory instability may be caused by dynamic characters of the distributed generators[2].It is one of the limiting criteria for synchronous operation of distributed generators[3].

Apart from the dynamics of the distributed generator, large dynamic loads in the distribution system also play a key role in the small signal stability of the distribution system, it is observed that dynamics of induction machine has a significant contribution to the power system oscillations [4][5][6].As of now, most of the works on small signal stability are on transmission systems. But with distributed generation and large dynamic loads the small signal stability investigation of the distribution systems is gaining importance. In [7] the small signal stability of a renewable energy based distribution system is investigated using Eigen value sensitivity. In [8] the critical parameters of distributed generators and dynamic loads that affect the power system stability are given. The existing literatures mainly deal with either voltage stability or rotor angle stability taking one at a time [2][7][9].Most of the

literatures focused on Eigen value sensitivity for the small signal stability assessment[2][3][7][8][9]. In some other literatures it is investigated through time domain analysis [4][9].

On the other hand after assessment of stability proper controllers are to be designed for the enhancement of the stability. In [11] controllability and observability indices are used for designing a damping controller. In [10] the comparison of PSS, SVC, STATCOM controllers for damping power system oscillations is given. However model matching techniques are well suited in these situations. They reduce the complexity in the design of the controller. In [11] the exact model matching of linear time varying multivariable systems is given. The exact model matching of the linear discrete time systems by periodic state or feedback law is given in [12][13].

The aim of the paper is investigation of the small signal stability of a distribution system considering both voltage stability and rotor angle stability of the distributed generator at a time. The various oscillatory modes in the system are observed using Eigen values. The critical modes are identified based on modes having least damping .The transfer function of the controllers that enhance the small signal stability are found using exact model matching. The test system is taken as a single machine infinite bus system with induction motor as load.

The rest of the paper is organized as follows: In section II, the mathematical modeling of the distribution system is given. In section III, the linearization of the whole system is given. In section IV the key investigations in small signal stability assessment are presented. Section V provides approach for controller design using exact model matching technique. Finally, the conclusion drawn from the results are summarized in section VI.

II MATHEMATICAL MODELS

The test system taken is a single machine infinite bus system with induction motor load. In this model, the power is supplied to the load (PL=1500MW,QL=150MVAR) from the infinite bus which is a distribution substation and local DG

unit (approximately PG=300MW,QG=225MVAR). The load at bus 2 is made of three parts: (i) one part represented by constant impedance load, (ii) another part represented by an equivalent large induction motor and (iii) a shunt capacitor for compensation purposes. The major portion of these loads is induction motor i.e. P=1485MW and Q=15 MVAR. The parameters of the induction motor, DG unit, transformers and transmission lines are given in Appendix.

With some typical assumptions the synchronous generator can be modeled by the following set of nonlinear differential equations [14]:

Mechanical equations:

Tdom edm edm ( X X )iqm Tdomss eqm

vds jvqs (Rs jX )(idm jiqm ) j(eqm jedm )

Where X' = Xs +Xm Xr /(Xm + Xr), is the transient reactance, X = X s+ X m, is the rotor open-circuit reactance, Tdom = (Lr + Lm) / Rr, is the transient open-circuit time constant, Te is the electrical torque, s is the slip, edm is the direct-axis transient voltage, eqm is the quadrature axis transient voltage, TL is the load torque, Xs is the stator reactance, Xm is the magnetizing reactance, Rs is the stator resistance, Hm is the inertia constant of the motor, Vds is the d-

s ( 1)

(1)

axis stator voltage, Vq s is the q-axis stator voltage, idm and iqm are d- and q-axis components of stator current respectively.

• 1 [P E ( X X )I I

  D]

  2H m q d q d q

  Here, this model represents the induction machine in its own direct and quadrature axes, which is different from the d-

  Generator electrical dynamics:

  .. (2)

  and q-axes of the synchronous generator. Therefore, axes transformation is used to represent the dynamic elements of

• 1

both the induction motor and synchronous generator with

q

E

Tdo

[K A (Vref Vo ) Eq ( X d X d )Id ]

(3)

respect to the same reference frame and to do so we use the following relations:

1

qm

0

V [V

V ]

Em

(edm

)2 (e )2

T

t 0

r (4)

tan1( edm )

e

m

qm

Where is the power angle of the generator, is the per unit rotor speed with respect to the synchronous reference, s

(Idm jI

qm ) (idm ji

jn

)e

qm

is the synchronous speed, H is the inertia constant of the generator, Pm is the mechanical input power to the generator

Vd jVq (vd jvq

)e jm

which is assumed to be constant, D is the damping constant of the generator, E q is the quadrature axis transient voltage, KA is the gain of the exciter amplifier, V ref is thereference voltage, V0 is the output of terminal voltage transducer, T do is the direct axis open circuit transient time constant of the generator, T r is the time constant of the terminal voltage regulator, X d is the direct axis synchronous reactance , X d is the direct axis transient reactance, Id and Iq are the direct axis,

Tm TL

Here the negative sign with idm and iqm indicates that they are opposite to Idm and Iqm when expressed in the same reference frame with synchronous generator. With this relation, a modified third-order induction machine model can be rewritten as:

quadrature axis currents of generator respectively and V t is the terminal voltage of the generator which is given by :

(Vd

• jVq

  ) (Rs

• jX )(Idm

• jI

  qm ) jEqm

  V [(E X I )2 ( X I )2

  1

  s [Tm

  • Em I

    qm ]

    t q d d d q

    2H

    • 1

      .. (5)

      A simplified transient model of a single cage induction machine with the stator transients neglected and rotor currents eliminated, is described by the following algebraic-differential

      Em

      T

      dom

      [Em

• ( X X )I dm ]

  ( X X )

  . (6)

  equations written in a synchronously-rotating reference frame [15]:

  m ss s

  T E

  dom m

  I qm

  .. (7)

  To complete the model, the d- and q-axis components of

  • 1 currents for both the generator and motor are given by the

    s 2H [Te TL ]

    following network interface equations:

    Tdom eqm eqm ( X X )idm Tdomss edm

    I n [(B cos G sin )E (B

    sin

• G cos

  )E ]

  di

  j1

  ij

  ij

  ij

  ij

  qj

  ij

  ij

  ij

  ij

  dj

  (8)

  I [(B

  n

  qi j1 ij

  sin

  ij Gij cos

  ij )Eq'j (Bij

  cos

  ij Gij sin

  ij )Ed'j ]

  Idm, Iqm are the linearized form of currents obtained in equations (8) and (9).

  .. (9)

  where ji = (j – i), Edj= 0, parameters Gij and Bij are the real and imaginary parts of the equivalent transfer impedances of the reduced network between ith and jth bus of the test system shown in Fig. 1.

  1. KEY INVESTIGATIONS IN SMALL SIGNAL STABILITY ASSESSMENT

     From the linearized equations (10)-(16) the linearized model of the whole system considering interconnection dynamics can be represented in the form of linear state space equations:

     Fig. 1Test System

     III LINEARIZATION OF DISTRIBUTION SYSTEM

     MODEL

     The power system model with dynamic load is expressed by equation (1)-(7). Using Taylor series expansion method and truncating the higher order terms, the linearized form of the equations (1)-(7) can be written as follows:

     x Ax Bu

     y Cx Du

     Where A is the state matrix, B is the input matrix, C is the control matrix and D is the output matrix. x are the states of the system. The states are , , Eq, V0, s, Em, m..

     u is the input disturbance. Vref is taken as the input disturbance. y is the change in the output for the input disturbance. As the complete small signal stability assessment requires investigation of both voltage and rotor angle stabilities the system is taken as single Input two output system. The outputs are and Vt.

     1. Eigen values and participation factors of the system

        Using the distribution system parameters given in the appendix and the values at the operating point obtained from

        the load flow analysis the state matrix A of the system can be calculated. The Eigen values are the roots of the characteristic

        0

        1

        s 0

        . (10)

        equation of the state matrix A .The Eigen values and their

        2H [Pm Eq Iq IqoEq (Xd Xq )Id Iq 0

        (Xd Xq )Id 0Iq D]

        (11)

        dampings and frequency of oscillations are given in table I. From the table we can observe that mode 6 and mode 7 has the least damping of 0.235.Hence they cause sustained oscillations in the system in the case of a disturbance and leads to instability. Hence these two modes are taken as the critical

        modes.

        Eq

        1 [K

        Tdo

        A (Vref

        • Vo

          ) Eq ( X d

        • X d )I d ]

        Table I. Eigen modes and their dampings

        1

        (12)

        T

        V0

        r

        [Vt V0 ]

        Eigen mode	value	damping	Freq(rad/s)
        1	-14.5	1	14.5
        2	-5.23+j9.54	0.48	10.9
        3	-5.23-j9.54	0.48	10.9
        4	-0.004+j0.19	0.23	0.20
        5	-0.004-j0.19	0.23	0.20
        6	-0.277+j0.29	0.68	0.48
        7	-0.277-j0.29	0.68	0.48

        (13)

        1

        s 2H [Tm Em I qm0 Em0 I qm ]

        (14)

        T

        1

        Em [Em ( X X )I dm ]

        dom

        (15)

        ( X X ) ( X X )

        m ss 0 T E

        I qm T

        E 2 I qm0 Em

        dom m 0

        dom m 0

        The contribution of states on oscillations was observed

        (16)

        here the suffix 0 denotes the value at initial operating point which can be obtained from the load flow analysis .Id, Iq,

        by evaluating the participation factors of each state on a particular mode. Participation factors gives the relationship among the states and Eigen mode [1].The normalized form of

        participation of the kth state in the ith Eigen mode can be given by:

        terminal voltage and speed of the DG unit have changed from the nominal values and settled at new values given by:

        Pki

        ki ki

        n

        ki ki

        1

        Vt ,new

        new

        Vt 0

        0

• Vt

Where ki is the Kth entry of right Eigen vector . ki is the kth entry of left Eigen vector i and n is number of state variables. The participation factors for the critical modes 6,7 are shown in tableII.From the table we can observe that deviation in rotor angle of DG unit and deviation in slip of induction motor are the dominant states that are contributing the oscillations in the critical modes.

Table II. Participation factors for critical modes

state	mode4	mode5
	0.1974	0.1974
	0.0951	0.0951
Eq	0.0212	0.0212
V0	0.0008	0.0008
s	0.2728	0.2728
Em	0.4086	0.4086
m	0.0041	0.0041

1. open loop response of the system

Fig.2 shows the open loop of the distribution system model shown in fig.1.It is a single input; two output system.G11(s) represents the open loop transfer function for the output Vt and input Vref. G21(s) is the open loop transfer function for the output and input Vref. The transfer functions are obtained through:

y(s) C(sI A)1B D

u(s)

Fig 2 Open loop of the system

The open loop response of Vt and for a step disturbance of Vref are shown in Fig4 and Fig5.From Fig4 and Fig5 we can observe that the open loop response of Vt and are oscillatory. Also the steady state values of Vt and

are not zero, but settling above zero. This indicates that

Fig 4 open loop response of G11(s)

Fig 5 open loop response of G21(s)

System is stable when terminal voltage and rotor speed of the DG unit are kept constant. For this to happen Vt and should be zero. Otherwise change in terminal voltage may lead to circulating currents in the distribution system and change in speed of the rotor of DG unit will lead to frequency deviations. Hence a control system is to be designed for the enhancement of the small signal stability which is given in the next section

1. DESIGN OF CONTROLLER USING EXACT MODEL MATCHING

   Controller for the enhancement of small signal stability is designed using exact model matching.

   1. meaning of exact model matching:

      The exact model matching design problem consists of determining a control law such that the input-output description of the closed loop system with controller is equal to that of a desired model and the orders of closed loop system and the desired model are strictly equal. The solution of this problem is of obvious practical importance, since it makes it possible to adapt the performance of a given system to that of a desired model.

   2. Control law

      The control part for the enhancement of the small signal stability is shown in Fig3.c1(s) and c2(s) are the transfer functions of the controllers which are to be determined using

      exact model matching. The closed loop transfer functions are given as:

      c1N (s)G21N (s)

      c (s)G (s)

      M (s)

      G (s)

      c1 (s)G11(s)

      Vt

      1D 21D

      (s) 21N

      11C

      1 c (s)G

      (s) c (s)c

      (s)G

      (s) V

      1 c1N (s)G11N (s) c1N (s)c2 N (s)G21N

      M 21D (s)

      1 11

      1 2 21

      (17)

      ref

      c1D

      (s)G11D

      (s)

      c1D

      (s)c2 D

      (s)G21D

      (s)

      G21C

      (s)

      c1 (s)G21(s)

      1 c (s)G (s) c (s)c

      (s)G

      (s)

      V

      . (22)

      1 11

      1 2 21

      ref

      Where

      .. (18)

      Fig.3 closed loop of the system with controllers

      c1N (s) and c1D(s) are numerator and denominator of c1(s). c2N (s) and c2D(s) are numerator and denominator of c2(s).

      G11N(s) and G11D(s) are numerator and denominator of G11(s). G21N(s) and G21D(s) are numerator and denominator of G21(s). M11N(s) and M11D(s) are numerator and denominator of M11(s). M21N(s) and M21D(s) are numerator and denominator of M21(s)

      Also G11D(s) = G21D(s) = GD(s) M11D(s) = M21D(s) = MD(s)

      The numerator of model function M11N(s) is taken as c1N(s) c2D(s) G11N(s) and M21N(s) is taken as c1N(s)c2D(s)G21N(s).From these approximations the equations(21) and (22) are reduced to a single equation

      M D (s) c1D (s)c2D (s)GD (s) c1N (s)G1N (s) c1N (s)c2 N (s)G21N (s)

      (23)

      MD is taken as 13th order function. For the exact model

      Now two model functions M11(s) and M21(s) which

      matching the orders on both sides of equation (23) should be same. Hence the controller transfer functions are of the form:

      indicates the desired performance of closed loop transfer

      functions are taken. The control law is defined as:

      c (s) c1N (s) s

      1 c (s) s6 a s5 b s 4 c s3 d s 2 e s f

      G11C (s) M11(s)

      . (19)

      1D 1 1 1

      1 1 1

      G21C (s) M21(s) (20)

      From equations (17) and (19) we can write

      (24)

      2

      c (s) c2N (s) (a s6 b s5 c s4 d s3 e s2 f s g )

      c2D

      (s) 2 2 2

      2 2 2 2

      c1N (s)G11N (s)

      c1D (s)G11D (s)

      1 c1N (s)G11N (s) c1N (s)c2 N (s)G21N (s)

      M11N (s)

      M11D (s)

      .(25)

      c1N(s) ,c1D(s) from (24) and c2N(s),c2D(s) from (25)are placed in the equation (23) .The equations then obtained are solved by

      c1D

      (s)G11D

      (s)

      c1D

      (s)c2 D

      1. G21D

         (s)

         equating the equal powers of s on both sides of the equation (23).Thus the controller transfer functions are obtained. The controller transfer functions c1(s) and c2(s) are then placed in equations (21) and (22) to obtain the closed loop response of

         . (21)

         And from equations (18) and (20) we can write

         Vt and for a step disturbance in Vref. The responses are shown in Fig6 and Fig7.From Fig6 and Fig7 we can observe that the closed loop response G11c(s) and G21c(s) are exactly matching desired responses M11(s) and M21(s).

         Vinf = 1.0

         Induction motor parameters:

         Rs = 0, Xs= 0.1, Xr= 0.18, Rr= 0.18, Xm = 3.2, Hm = 1.5s.

         ACKNOWLEDGMENT

         The authors would like to thank P.Preeth, T.K.Sindhu, R.Sunitha, K.Sunitha and S.Kumarvel from National institute of Technology for their valuable technical assistance.

         REFERENCES

         Fig 6 closed loop response and model response of Vt

         Fig 7 closed loop response and model response of

2. CONCLUSION

This paper has presented a systematic approach for assessment and enhancement of small signal stability of DG based distribution system. The critical oscillatory modes in the system are identified with the help of Eigen values. The open loop response of the change in terminal voltage and deviation in rotor speed of DG unit for a disturbance in the field circuit is assessed. The desired models M11(s) and M21(s) are chosen by providing sufficient damping to the Eigen values of models. Finally the controllers c1(s) and c2(s) are designed using exact model matching to ensure that the closed loop response of Vt and for a disturbance in Vref is having oscillatory stability and also ensured that they match the model functions M11(s) and M21(s).The controller transfer functions obtained are of higher orders i.e. sixth order. Physical implementation of such higher order controllers is a difficult task. The desired response can be obtained with lower order controllers also. But in that case the orders of closed loop transfer function and model fiction may or may not equal. Also the closed loop response will approximately follow desired response. Using optimization techniques we can design the controllers such that the error between actual and desired response will be minimum. That is called approximate model matching which will be the future work.

APPENDIX POWER SYSTEM PARAMETERS

The parameters used for the SMIB system are given below: Synchronous generator parameters:

Xd= 2.1, Xd= 0.4, H = 3.5s, Tdo= 8, D = 4.

Automatic Voltage Regulator (AVR) Parameters: KA = 50, Tr = 0.1.

Transformer Parameter: XT = 0.016.

Transmission Line Parameters: Xe = 0.027.

Infinite bus:

1. P. Kundur, Power System Stability and Control, Electric Power Research Institute.

2. N.K.Roy,H.R.Pota, T.F.Orchi,small signal stability assessment of active distribution networks with dynamic loads, in Proc. IEEE power engineering conference,2012,pp.1-7

3. L. Dong-Jing and W. Li, Small-signal stability analysis of an autonomous hybrid renewable energy power generation/energy storage system Part I: Time-domain simulations, IEEE Trans. Energy Conservation., vol. 23, no. 1, pp. 311320, Mar. 2008.

4. B. M. Nomikos and C. D. Voumas, "Investigation of induction machine contribution to power system oscillations," IEEE Trans. On Power Systems, vol. 20, no. 2, pp. 916-925, 2005.

5. J. V. Milanovic and 1. A. Hiskens, "Effects of dynamic load model parameters on damping of oscillations," Electric Power System Research, vol. 33, pp. 53-61, 1995.

6. A. Borghetti,R. Caldon, A .Mari and C. A. Nucci, "On dynamic load models for voltage stablities studies," IEEE Trans. on Power Systems, vol. 12, no. I, pp. 293-303, 1997

7. S.Dahal,N.mithulananthan and Tapan kumar Saha Assessment and enhancement of small signal stability of a renewable energy based electricity distribution system IEEE Trans. On Sustainable Energy, vol.3, No .3, July 2012

8. M. A. Mahmud, M. J. Hossain, and H. R. Pota, "Investigation of critical parameters for power system stability with dynamic loads," in IEEE PES General Meeting, Minneapolis, MN, USA, July 2010, pp. 1-6.

9. S. Dahal, N. Mithulananthan, and T. Saha, Investigation of small signal stability of a renewable energy based electricity distribution system, in Proc. IEEE Power and Energy Society General Meeting,2010, pp. 18.

10. N. Mithulananthan, C. A. Canizares, J. Reeve, and G. J. Rogers,

Comparison of PSS, SVC, and STATCOM controllers for damping power system oscillations, IEEE Trans. Power Syst., vol. 18, no. 2,

pp. 786792, May 2003.

[11]T.K. Sunil Kumar, Model Matching Controller design methods with applications in Electrical Power Systems Ph.D thesis, IIT Kharagpur 2009.

1. S.H.Wang, C.A.Desoer, The Exact Model Matching of Linear Multivariable Systems , IEEEtransactions on Automatic control,

   JUNE 1952

2. David.M.Alter,Tsu chin Tsao,Two dimensional exact model matching with application to repetitive controlASME transactions on dynamic systems, measurements,control,vol 116,March 1994.

3. Jan Machowski, Janusz W. Bialek, James R. Bumby, power system dynamics and stability Wiley publications

4. C.W. Taylor , power system voltage stabilityMcGraw- hill editions.