Stability Analysis on Effect of Jawa Bali Power System Loading Using Developed State Space Program - repository civitas UGM

  δ is the angular position of the rotor ω is the angular velocity of rotor

  ′ is the q-axis transient synchronous reactance

  I

  − ) ₂ ⁽

  ) (

  π δ Q D ^{j j q d} jV

• = +

  V Ve e jV V + = = + − θ

  π δ ) ₂ ⁽

  ) ( Figure 2 Two-axis synchronous machine model

  Defines :

  X d is the d-axis transient synchronous reactance X q

  is the armature resistance

  Rs

  I X

  I d is the d-axis current refer to the mach. ref. frame I q is the q-axis current refer to the mach. ref. frame

  I D is the d-axis current refer to the netw. ref. frame

  IQ is the q-axis current refer to the netw. ref. Frame

  τ q′0 is the q-axis transient open circuit time constant τ d′0 is the d-axis transient open circuit time constant

  X d is the d-axis synchronous reactance X q is the q-axis synchronous reactance

  Eq ′ is the q-axis transient voltage Ed ′ is the d-axis transient voltage E fd is the DC field voltage

  ωs is the rated value of an angular velocity of rotor

  TM is the mechanical torque applied Te is the electromagnetic torque developed TD is the damping torque component D is the damping factor H is the inertia constant

  From the above figure, the stator algebraic equation can be written as : )

  X E ^{Q D j q d} jI I e jI

  E e j

• ′ ^j _{q q d q d}

  ) )( (

  ሶ (1) where the matrices A and B also depend on the given operating conditions.

  

Stability Analysis on Effect of Jawa-Bali Power System Loading

Using Developed State Space Program

  Avrin Nur Widiastuti Electrical Dept., Engineering Fac. Gadjah Mada University, e-mail:avrin@.ugm.ac.id, pinkpisan@yahoo.com

  Abstract – This study develops a program to generate linearized state space representation of the power systems. The two-axis model of synchronous generator and the IEEE type I model of exciter are used to represent dynamic behaviors of the generating units. The developed program can accommodate an actual structure of the power systems with some numbers of generating units and the AC buses. Furthermore, the effect of loading on Jawa Bali Power System is studied. The real loads at a particular bus are increased continuously. At each step, the initial conditions of the state variables are computed, after running the load flow, and linearization of the equations is done. The matrix A is formed, and its eigenvalues are checked for stability. If unstable, the program will identify the states associated with the unstable eigenvalues of matrix A using the participation factor method. From the result of the developed program, we see that the dynamic performance of the system is change as the given operating conditions change, the real part of eigenvalue is bigger so that the system is more unstable. All of the unstable modes are associated with state variable

• ′

  δ of machine 1, 6 and 8, and also the state variable q

  E of machine 5.

  With this study, it helps Indonesian utility to predict the maximum loading of the powers system within stability region and for control design, thereafter.

I. I NTRODUCTION

  The analysis of dynamic stability can be performed by deriving the linearized state space model of the system in the following form : ݔ ൌ ܣݔ ൅ ܤݑ

  The eigenvalues of the state matrix A determine the small-signal stability around the given operating point. The eigenvalues analysis can be used not only for the determination of the stability regions, but also for the design of the controllers [1].

  − ′

  In this study we develop the program to generate linearized state space model of multimachine power systems that is necessary for the study of both small signal stability analysis, voltage stability and for control design, thereafter.

  I X j R Ve ] ) ( [

  j q d d s j e jI

  θ −

  π δ

  X j ′ _s R [ ] ^{) 2 (}

  ) (

  π δ −

  ′

  2 (

• ′

II. POWER SYSTEM DYNAMIC MODEL

• synchronous generator and the associated excitation systems
• AC transmission network including static fixed-impedance, loads For system stability studies concerning electromechanical oscillation, it is appropriate to neglect the transmission network and the machine stator transients [2]. The dynamics of synchronous generator rotor circuits and excitation systems are represented by the sets of differential equations. The result is that the complete system model consists of a large number of non- linear ordinary differential and algebraic equations. We can describe complex power system as our plant which consists of generators, transmission network as depicted in Fig. 1. It is noted that there can be a number of generators in the system.

  F τ

  is the voltage regulator time constant

  E K

  is the exciter gain

  E τ

  is the exciter time constant

  F K

  is the regulator stabilizing circuit gain

  is the regulator stabilizing circ time constant

  is the voltage regulator gain

  EX S

  is the rotating exctr saturation at ceiling voltage

  EX A

  is the saturation const for rotating exciters

  EX B

  is the saturation const for rotating exciters

AUN/SEED-Net RC-EEE / ISMAC 2013 February 4-5, 2013, Bangkok, Thailand.

  The overall power system representation in this research includes models for the following components:

  A τ

  We will use a simplified generator model with two axes adopted from [3] as show in Fig. 2 ^d

  Synchronous Generator Model

  q q d q d E e j

  Figure 1 Complex power system A.

  Network including load generator generator

  )

  2 (

  = ′ + ′ − ′ + ′ −

  − π

  δ j

  I X

  is the stabilizer voltage

  X E (2) fd

  E

  is the exciter voltage

  R

  V

  is the regulator voltage

  F R

  A K

  AUN/SEED-Net RC-EEE / ISMAC 2013 February 4-5, 2013, Bangkok, Thailand.

  π generators and called generator buses. The rest of the buses which are not

  ( ) − j δ −

  connected to generating units called as load bus.

  2 e

  Generator buses parts, we obtain : The two equations representing real and reactive power balance for each generator bus are given below. For real power balance,

• Multiplying eq. (2) by and equating the real and the imaginary

  ′ ′

  E − V sin( δ − θ ) − R

  I X I = d s d q q sin( − + ) cos( − )

• (3)

  I V δ θ

  I V δ θ

  E ′ V cos( δ θ ) R

  I X ′ I (4) − − − = q s q d d n

• di i i i qi i i i

  In addition, the dynamic behaviors of the rotor motion and the rotor circuits (16)

  −

  V V Y cos( θ − θ − α ) =

  can be describing using the following differential equations : i k ik i k ik

  ∑ k

  1 =

  & δ = ω − ω s

  (5) For reactive power balance,

  T − T ) M e D

  I V cos( δ − θ )

• ( T −

  I V sin( δ − θ ) di i i i qi i i i

  & = ω

  M

  (6) n

  ′ ′ E − ( X − X ) I − E −

  V V Y sin( θ − θ − α ) = (17) ( ) fd d d d q i k ik i k ik

  & ∑

  ′ E = q k

  1 =

  (7)

  ′ τ d

  for

  i = 1 , 2 ,..., m ′ ′ (

  X − X ) I − E ( q q q d )

  & ′

  E = d

• Load buses

  (8)

  τ ′ For the load buses, we have equations for both real and reactive power also. q

  For real power balance, where

  n Δ

  2 H

  (18)

  = P ( V ) −

  V V Y cos( θ − θ − α ) = M Li i i k ik i k ik

  ∑

  (9)

  ω s k =

  1 For reactive power balance, T = E ′

  I I e ( q q d d q d d q )

  I E ′ + + I ( X ′ − X ′ )

  (10)

  n T = D ( )

  ω − ω

  (19)

  D s Q ( V ) −

  V V Y sin( θ − θ − α ) = Li i i k ik i k ik

  (11) ∑

  k =

  1 Excitation System B.

  For

  i = + m 1 ,... n

  In order to improve the damping characteristics of the synchronous

  P and Q and are voltage dependence loads. In this study, loads are Li Li

  generator, supplementary excitation controls have been widely employed. In assumed as constant power type. this study, we use the IEEE type I exciter as shown in Figure 3

  V _ref

• D. Linearized State Space Model of Power Systems
• − ⎡ ⎤ ⎡ ⎤

  K A

  V R

  1 E fd

  V ⎢ ⎥ ⎢ ⎥

• 1 s K s

  τ τ ⎣ A ⎦ ⎣ E E ⎦

  Linearized State Space Model −

• −

  The behavior of a dynamic system, such as a power system, may be

  ( ) S E E fd

  described by a set of n first order nonlinear ordinary differential equations of the following form :

  & = ( u , ) (20) x f x

  ⎡ ⎤ sK F

  ⎢ ⎥

  If equations (20) is linearized around the given operating states and inputs +

  1 s τ ⎣ F ⎦ ( x , u ) , then the linearized state equation can be written as follows :

  Figure 3 Block diagram of the IEEE type I excitation control

  Δ& x = A Δ x B Δ u (21) +

  where From the block diagram above, we have three following equations :

  ⎡ ⎤

  ∂ f ∂ f ⎡ ∂ f ∂ f ⎤

• K S ( E )

  1

  1

  1

  1 V E E fd

  & ... ...

  (12)

  E = E ⎢ ⎥ ⎢ ⎥ fd fd

• R

  ∂ ∂ ∂ x x u u T

  1 n r τ

  E E ⎢ ⎥ ⎥ ⎢ A = ... ... ... B = ... ... ...

  ⎢ ⎥ ⎢ ⎥ −

  V K K K K R A A F A &

  ∂ f ∂ f ∂ f ∂ f

  V = + R − E ( V − V ) ⎢ ⎥ ⎢ ⎥ R f fd ref

• (13) n n n n

  τ τ τ τ τ A A A F A

  ⎢ ⎥ ⎢ ⎥ ∂ x ∂ x ∂ u ∂ u 1 n 1 r

  ⎣ ⎦ ⎣ ⎦

  (22)

  Rfi K &

  Reference [4] shows that the linearized state space model will be usefull (14)

  R = − E fi fdi

• Fi

  2 for design controller that can be applied in the system.

  τ i ( τ )

  F Fi

• Eigenvalue and Stability where

  After the linearized state space representation of the system is obtained

  B × E _{EX fd}

  such as eq.21, we can see the eigenvalues of the system by inspection of

  S A e (15) = ⋅

  E EX matrix A , which reveal the stability of system.

  A negative real eigenvalue represents a decaying mode. The larger its Defines : magnitude, the faster the decay (system stable), while a positive real eigenvalue represent instability [5,6,7,8,9].

  For a complex pair of eigenvalues = ± j the frequency of

  λ σ ω

  C. AC Transmission System

  oscillation in Hz is given by : The network characteristics in this study described in the form of power balance. The network can consist of many electrical buses and branches

  ω

  depending on the system of interest. Some of the buses connect to the f (23)

  = 2 π

  

AUN/SEED-Net RC-EEE / ISMAC 2013 February 4-5, 2013, Bangkok, Thailand.

  Figure 4 An algorithm to generate linearized state space model In addition, the damping ratio can be computed as :

  Figure 5 Flowchart on effect of loading of power system (24)

  ζ =

  2

2 III.

• SIMULATION AND RESULT

  σ ω State Space Model Generation Algorithm A.

• Participation Factor The first step of the program is read input data from all components. After Participation factor analysis aids in the identification of how each state that the program will run load flow to calculate the operating point of the variable is reflected on a given mode or eigenvalue [3]. Participation factor network. From operating point, program will continue to calculate initial is defined as

  p

  condition all variables in generator. Then program will calculate the matrix

  ki

  for each equation and do many calculations based on the algorithm to develop

  v w ki ik state model [10]. The flow chart is in Fig. 4.

  (25)

  p = ki n v w

  ∑ 1 ki ik k =

  Where

  Effect of loading on 500 KV Jawa Bali power system B. th th

  In this study the effect of loading on the small signal stability of the

  p is the participation factor relating the k state variable to the i ki

  system is studied using the developed program. The real or reactive loads at a eigenvalue particular bus/buses are increased continuously. At each step, the initial

  th

  conditions of the state variables are computed, after running the load flow, and are the entries in the left and right eigenvector

  w v k ki ki

  and linearization of the equations is done. The matrix Asys is formed, and its

  th

  associated with the eigenvalue. eigenvalues are checked for stability. If unstable, the program will identify

  i

  the states associated with the unstable eigenvalues of matrix Asys using the A further normalization can be done by making the largest of the participation factor method. The flow chart of this study is in Figure 5. participation factors equal to unity.

  The developed program is applied for 500 KV Jawa-Bali power system in The formulation of the state equations involves the development of

  Indonesia. In Figure 6 we can see graphical structure of the 500 KV Jawa- linearized equation about an operating point and elimination of all non state Bali power system. and non input variables. However, the need to allow for the representation of

  This system consists of eight generators and 24 buses. Generator in extensive transmission network, loads and excitation system, makes the Suralaya (no 1) is a slack/swing bus, no 2- 8 are a PV bus (see Appendix). process very complex. Therefore, the formulation of the state equations The rest of buses are load buses. requires a systematic procedure for treating the wide range of devices.

  3 4 2 3.39 8 0.06 0.69 0.93

  3 6 2 3.95 9.45 1.5 2.01 2.12 2.01

  0.48

  7

  0.29

  2.19

  2 8 2 3.45 7.9 0.47 2

  7

  0.29

  1 7 2 3.95 9.45 1.5 2.01 2.12 2.01

  7

  0.29

  4

  3 Bus

  0.60

  9

  0.25

  9

  1.63

  3 5 0 2.71 4.57 0.5 1.381

  5

  0.24

  2

  0.30

  7

  No A B Ke tE kA tA kF tF

  3

  0.67

  40

  0.67

  0.1 0.12 1; 8 0.09 0.368 -0.06

  40

  0.67

  0.1 0.12 1; 7 0.09 0.368 -0.06

  40

  0.67

  0.1 0.12 1; 6 0.09 0.368 -0.06

  40

  0.1 0.09 1; 5 0.09 0.368 -0.06

  1 0.09 0.368 -0.06

  40

  0.9

  0.1 0.09 1' 4 0.12 0.535 -0.08

  40

  0.9

  0.1 0.12 1; 3 0.12 0.535 -0.08

  40

  0.67

  0.1 0.12 1; 2 0.09 0.368 -0.06

  40

  0.67

• 10
• 8
• 6
• 4
• 2
• 15 -10 -5

  0.27

  Figure 6. Graphical structure of 500KV Jawa Bali power system

  8 PITON7 PITON 3254 fossil steam PV Table 2 Parameters of Generator data

  5

  2 37 0.9466+0.0000i

  1 , δ δ

  8

  1 24 4.2152+0.0000i

  No No of eigenvalue/mode Critical eigenvalue(s) Participation factor

  Figure 7 Plot of eigenvalues on the base case Table 4 Critical eigenvalues and the their participation factors on the base case

  Table 3 Parameters of the generator excitation systems .

  This research concern with small signal stability study, specially the effect of loading level in the system. The real loads at a particular bus is increased continuously. For this study we will increase bus no 13 which is DEPOK bus (see Bus Data in Appendix).

  The input data needed to run the program are : network data and parameter data of generator and exciter. Network data is consists of bus data, generator data, and branch data. The network data used for the study is given in the Appendix . From Figure 6, we can see that there are 8 generators. Table 1 gives the detailed capacity and type of them. Also the parameters of the generator and exciter are given in Table 2 and Table 3.

  PV

  E δ

  7 GRSIK7 GRESIK 1050 combine cycle

  PV

  6 GRATI7 GRATI 527 combine cycle

  5 TJATI7 TANJUNG JATI 1320 fossil steam PV

  4 SGLNG7 SAGULING 700 Hydro PV

  3 CRATA7 CIRATA 1008 Hydro PV

  PV

  2 MTWAR7 MUARA TAWAR 2941 combine cycle

  1 SLAYA7 SURALAYA 3400 fossil steam Slack

  No Bus Code Bus Name Capacity (MW) Type of gen Type of bus

  Table 1 Generator capacity data

  1 , q

  3 40 0.2636+0.0000i

  4

  Xq Xd d

  0.27

  1 0.573 0.88

  0.07

  7 3 2 2.4 9.99

  0.29

  1 2 2.64 5.69 1.5 2.02 2.11 0.28 0.49 3

  ି૜

  ૚૙

  X ′ Rs

  X ′ p

  τ ′ q τ ′

  6 δ

  Bus D H d

  10 real axis im agi nar y ax is

  8

  6

  4

  2

  10

  5

  Table 5 Critical eigenvalues and the associated state

  We plot in the range real axis from -15 to 10, and imaginary axis from -10 to 10. It is because we just concern on the positive real axis. For result of eigenvalue above, we see that there are 3 modes which has positive real part. It means those modes are unstable. For the participation factor, we have matrix 56 by 56, and we can check associated states which give big contribution to those modes. Only participation factor greater then 0.3 are listed. Those modes and their participation factor are listed in Table 4.

  In the first time, DEPOK is loading on base case which has real load 422 KW and reactive load 174 KVAR. After running load flow, the program will calculate initial condition, then the program will generate eigenvalues of the system. The number of generator is 8, each generator has 7 states, so that we have 56 eigenvalues . Figure 7 shows the plot of eigenvalues on base case.

  0.1 0.12 1; AUN/SEED-Net RC-EEE / ISMAC 2013 February 4-5, 2013, Bangkok, Thailand. Some notable observations regarding small-signal dynamic behaviors of the 500KV Jawa Bali power system include :

• The calculation of participation factor is very useful to identify how each dynamic variable affects a given mode or eigenvalue
• We see that after we change loading level, the dynamic performance of the system is change as the given operating conditions change, the real part of eigenvalue is bigger so that the system is more unstable.
• All of the unstable modes are associated with state variable
• Using participation factor we can onstruct reduced-order models for dynamic stability studies by retaining only a few modes as to which state variables significantly participate in the selected modes.

  8

  For the base case, we see that the system is not stable due to electromechanical variable

  6 δ

  41 0,2809

  E

  5 , δ q

  8

  38 1,2268

  δ δ

  1 ,

  20 10,2268-6,1229i

  of machine 1, 6 and 8. Also the state variable

  1 , δ δ

  8

  19 10,2268+6,1229i

  4220MW (10xbase case)

  6 δ 7.

  40 0,2605+0.0000i

  E

  5 , δ q

  8

  36 1,1218+0.0000i

  δ

  q E of machine 5 contributes to the instability of the system.

  1 ,

  [2] Prabha Kundur, Power System Stability and Control. New York: McGraw-Hill, Inc, 1994. [3] Peter W. Sauer, M.A. PAI, Power System Dynamics and Stability, Prentice Hall, 1998. [4]

  2007

AUN/SEED-Net RC-EEE / ISMAC 2013 February 4-5, 2013, Bangkok, Thailand.

  Electronics Engineering Fieldwise Seminar 2007 , Bangkok, Thailand,

  Avrin Nur Widiastuti, “State Space Model Generation for Stability Analysis of Power Systems”, Proceeding of the Electrical and

  [8] William L. Brogan, Modern Control Theory, Prentice Hall International Edition, 1991. [9] Dr. David Banjerdpongchai, Lecture note of Control System Theory : Control System Theory , Control System Research Laboratory, 2002. [10]

  Katsuhiko Ogata, State Space Analysis of Control Systems, Prentice Hall International, Inc, 1967

  [7]

  Iowa State University Press, 1997

  [5] Katsuhiko Ogata, Modern Control Engineering, Prentice Hall International, Inc, 1997. [6] P.M. Anderson, A.A Fouad, Power System Control and Stability, The

  the 32nd Conference on Decision and Control , San Antonio, Texas, 1993.

  V.Vittal, M.H. Khammash, C.D. Pawloski, “Analysis of control performance for stability robustness of power system“, Proceeding of

  the dynamic stability analysis of power systems including HVDC links”, IEEE Transaction on Power System , 1981, P: 1871-1879.

  To see the effect of loading, we increase real load at bus 13. Start from base case (422KW), then we increase to 844 MW until 4220 MW (10 time of base case). This is the maximum loading of the scenario due to limitation of generating capacity.

  [1] K.R. Padiyar, M.A. Pai, C. Radhakrishna, “A versatile system model for

  R EFERENCES

  q E of machine 5.

  of machine 1, 6 and 8, and also the state variable

  δ

  Due to large size of the power system, it is often necessary to construct reduced-order models for dynamic stability studies by retaining only a few modes. The appropriate definition and determination as to which state variables significantly participate in the selected modes become very important. Therefore, this study is very useful to determine state variables which give significantly participate in the selected modes.

  q E of machine 5.

  of machine 1, 6 and 8, and also the state variable

  δ

  From Table 5 we observe that, for increased load at bus 13, the system become more unstable. The eigenvalues are moving to positive real axis. For the last scenario, there is one complex eigenvalue. All of the unstable modes are associated with state variable

  δ δ

  8

  on different loading at bus 13

  844 MW (2xbase case)

  24 6,0130+0.0000i

  1266 MW (3xbase case)

  δ 3.

  40 0,2631+0.0000i 6

  5 q E

  37 1,0403+0.0000i

  , δ δ

  1

  24 5,08894+0.0000i 8

  δ 2.

  1

  40 0,2636+0.0000i 6

  E δ

  1 , q

  5

  37 0,9466+0.0000i

  , δ δ

  1

  8

  1. 422 MW 24 4,2152+0.0000i

  Scenario Load at bus 13, DEPOK Critical eigenvalue Associated Mode states Eigenvalues

  8

  , δ δ

  23 8,6932+0.0000i

  2110 MW (5xbase case)

  2532MW (6xbase case)

  6 δ 6.

  40 0,2613+0.0000i

  E

  5 q

  36 1,1238+0.0000i

  δ δ

  1 ,

  8

  23 7,8396+0.0000i

  6 δ 5.

  35 1,0906+0.0000i

  40 0,2626+0.0000i

  5 q E

  35 1,0906+0.0000i

  1 , δ δ

  8

  24 6,0130+0.0000i

  1688 MW (4xbase case)

  6 δ 4.

  40 0,2626+0.0000i

  5 q E