Tuning of Power System Stabilizer for SMIB system using by miy51275

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									   A Novel approach for tuning Power System Stabilizer (SMIB
         system) using Genetic Local Search technique
      J.SATHEESHKUMAR JEGADEESAN                      Dr.A.EBENEZER JEYAKUMAR
        Department of Electrical Engineering          Department of Electrical Engineering
        Government College of Technology              Government College of Technology
              Coimbatore, INDIA                             Coimbatore, INDIA


Abstract: - The genetic local search technique (GLS) hybridizes the genetic algorithm (GA) and
the local search (such as hill climbing) in order to eliminate the disadvantages in GA. The
parameters of the PSS (gain, phase lead time constant) are tuned by considering the single
machine connected to infinite bus system (SMIB). Here PSS are used for damping low
frequency local mode of oscillations. Eigen value analysis shows that the proposed GLSPSS
based PSS have better performance compared with conventional and the Genetic algorithm
based PSS (GAPSS). Integral of time multiplied absolute value of error (ITAE) is taken as the
performance index of the selected system. Genetic and Evolutionary algorithm (GEA) toolbox is
used along with MATLAB/SIMULINK for simulation.

Key-words: - Genetic local search, Power system oscillations, Power system stabilizer, Genetic
algorithms, SMIB, K-constants
                                               power system utilities still prefer the
1. Introduction                                conventional lead-lag PSS structure. The
           Power systems experience low-       reasons behind that might be the ease of on-
frequency oscillations due to disturbances.    line tuning and the lack of assurance of the
These low frequency oscillations are           stability related to some adaptive or variable
related to the small signal stability of a     structure techniques.
power system. The phenomenon of stability                  Unlike     other       optimization
of synchronous machine under small             techniques, Genetic Algorithm (GA) works
perturbations is explored by examining the     with a population of strings that represent
case of a single machine connected to an       different potential solutions therefore, GA
infinite bus system (SMIB). The analysis of    has implicit parallelism that enables it to
SMIB [4] gives physical insight into the       search the problem space globally and the
problem of low frequency oscillations.         optima can be located more quickly when
These low frequency oscillations are           applied to complex optimization problem.
classified into local mode, inter area mode    Unfortunately, recent research has identified
and torsional mode of oscillations. The        some deficiencies in GA performance. This
SMIB system is predominant in local mode       degradation in efficiency is apparent in
low frequency oscillations.            These   applications with highly epistatic objective
oscillations may sustain and grow to cause     functions, i.e. where parameters being
system separation if no adequate damping       optimized are highly correlated. In addition,
is available. In recent years, modern control  the premature convergence of GA represents
theory have been applied to power system       a major problem.
stabilizer (PSS) design problems. These           In this proposed genetic local search (GLS)
include optimal control, adaptive control,     [7] approach, GA is hybridized with a local
variable structure control, and intelligent    search algorithm to enhance its capability of
control.                                       exploring the search space and overcome the
           Despite the potential of modern     premature convergence. The design problem
control techniques with different structures,
with mild constraints and an eigenvalue-                    K3                  K3 K 4
                                               Eq =
                                                 '
                                                                     E fd                    (3)
based objective function.                              1  sTd 0 K 3
                                                              '
                                                                             1  sTd' 0 K 3
           The GLS algorithm is employed
to solve this optimization problem and         Vt  K5   K 6 Eq
                                                                   '
                                                                                                   (4)
search for the optimal settings of PSS         The K-constants are given in appendix.
parameters. The proposed design approach       The system data are as follows [8]:
has been applied to SMIB system. Eigen         Machine (p.u):
value analysis and simulation results have                    '
                                               xd = 0.973 xd = 0.19
been carried out to assess the effectiveness
and robustness of the proposed GLSPSS to       x q = 0.55      Td' 0 = 7.765 s                    (5)
damp out the electromechanical modes of
oscillations and enhance the dynamic           D = 0.0          H = 4.63
stability of power systems.
                                               Transmission line (p.u):
                                               re = 0.0        xe = 0.997                         (6)
2. System investigated                         Exciter :
           A single machine-infinite bus
(SMIB) system is considered for the            KA = 50.00 TA= 0.05 s                              (7)
present     investigations.   A     machine    Operating point :
connected to a large system through a
transmission line may be reduced to a          Vt 0 = 1.0      P0 = 0.9                           (8)
SMIB system, by using Thevenin’s
equivalent of the transmission network         Q0 = 0.1       δ0 = 65 0
external to the machine. Because of the         The interaction between the speed and
relative size of the system to which the       voltage control equations of the machine is
machine is supplying power, the dynamics       expressed in terms of six constants k1-k6.
associated with machine will cause             These constants with the exception of k3,
virtually no change in the voltage and         which is only a function of the ratio of
frequency of the Thevenin’s voltage            impedance, are dependent upon the actual
(infinite bus voltage). The Thevenin           real and reactive power loading as well as the
equivalent impedance shall henceforth be       excitation levels in the machine.
referred to as equivalent impedance (i.e.      Conventional PSS comprising cascade
Re+jXe).                                       connected lead networks with generator
           The synchronous machine is          angular speed deviation () as input signal
described as the fourth order model. The       has been considered. Fig.1 shows the
two-axis        synchronous         machine    linearized model of the single machine
representation with a field circuit in the     connected to large system (SMIB) around the
direct axis but with out damper windings is    operating point.
considered for the analysis. The equations
describing the steady state operation of a
synchronous generator connected to an
infinite bus through an external reactance
can be linearized about any particular
operating point as follows(eq:1-4):
             d 2 
Tm  P  M                            (1)
              dt 2
P  K1  K2Eq  '
                                         (2)


                                                 Fig.1 Linearized model of SMIB system
                                                          contribute to the negative synchronizing
From the transfer function block diagram,                 torque component.
the following state variables are chosen for                          Wash out function (Tw) has the
single machine system. The linearized                     value of anywhere in the range of 1 to 20
differential equations can be written in the              seconds. The main considerations are that it
form state space form as                                  should be long enough to pass stabilizing
  0                                                       signals at the frequencies of interest relatively
X(t)  A.x(t)  BU(t)                      (9)            unchanged, but not so long that it leads to
                                                          undesirable generator voltage excursions as a
Where                                                     result of stabilizer action during system-
                                                          islanding conditions. For local mode of
X(t) = [ Δδ Δω ΔEq’ ΔEfd]T            (10)                oscillations in the range of 0.8 to 2 Hz, a
        0          314          0               0       wash out of 1.5 sec is satisfactory. From the
A=                                            0         view point of low-frequency interarea
    D M           K1 M     K2 M                   
        K4          0      1 K 3T 'do     1 T 'do    oscillations, a wash out time constant of 10
                                                     
    K A TA K 5         K A TA K 6
                      0                      1 TA        seconds or higher is desirable, since low-
                                                          time constants result in significant phase lead
                                                 (11)     at low frequencies.
             .            T                                          The stabilizer gain K has an
                                                         important effect on damping of rotor
                   KA                           (12)
 B 0 0 0                                               oscillations. The value of the gain is chosen
                  TA                                    by examining the effect for a wide range of
                                                          values. The damping increases with an
         System state matrix A is a                       increase in stabilizer gain upto a certain point
function of the system parameters,                        beyond which further increase in gain results
which depend on operating conditions.                     in a decrease in damping. Ideally, the
Control matrix B depends on system                        stabilizer gain should be set at a value
parameters only. Control signal U is the                  corresponding to maximum damping.
PSS output. From the operating                            However the gain is often limited by other
conditions and the corresponding                          considerations. The transfer function model
parameters of the system considered,                      of the SMIB system with the PSS is given in
the system eigenvalues are obtained.                      Fig.2.
                                                                     The transfer function of Power
                                                          System Stabilizer is given by,
                                                                            10s      1  sT1 2
3. Transfer function model of                              H1(s)  K *            *(          )       (13)
                                                                         (1  10s) 1  0.05 s
PSS and design considerations
          The exciter considered here is                  Where,
only having the gain of KA and the time                     K—PSS gain
constant of TA. The typical PSS consists of                 Tw—washout time constant
a washout function, a phase compensator                     T1-T4---phase lead time constants
(lead/lag functions),and a gain. It is well               A low value of T2=T4=0.05 sec is chosen
known that the performance of the PSS is                  from the consideration of physical
mostly affected by the phase compensator                  realization. Tw=10 sec is chosen in order to
and the gain. Therefore, these are the main               ensure that the phase shift and gain
focus of the tuning process. Two first order              contributed by the wash out block for the
phase compensation blocks are considered.                 range of oscillation frequencies normally
If the degree of compensation required is                 encountered is negligible. The wash out time
small, a single first-order block may be                  constant (Tw) is to prevent steady state
used. Generally slight under compensation                 voltage off sets as system frequency changes.
is preferable so that the PSS does not                    Considering two identical cascade connected
                                Fig.2 SMIB system with power system stabilizer


  lead-lag networks for the PSS T1=T3 .              Where
  Hence now the problem reduces to the              n is the undamped natural frequency of
  tuning of gain (K) and T1 only.                   the corresponding root.
                 The parameters of the PSS          For the determination of PSS parameters
  obtained for the damping ratio of 0.3. The        a damping factor of =0.3 is chosen
  oscillation frequency is generally about          (maximum damping). Corresponding to
  0.8-2 Hz for the local mode of oscillations.      this damping factor the desired
  In this SMIB system only local mode of            eigenvalues are obtained as.
  oscillations are considered for the tuning of
  PSS. The local mode of oscillation occurs         1= -n+ n  1   2
  when a machine supplies power to a load           2= -n- n  1   2
  center over long, weak transmission lines.
  Pole placement technique is used for the          It is to be noted here that the some of the
  tuning of CPSS design [6].                        eigenvalues need not be shifted since
                                                    they are placed for off in LHP. If any
  3.1. Conventional PSS design
                                                    electromechanical modes of oscillations
                 The eigenvalues of the above
                                                    are present then PSS needs to be added to
A matrix are obtained using Matlab. It is
                                                    enhance the dynamic stability of the
evident from the open loop eigenvalues, the
                                                    system. By using Decentralized modal
system without PSS is unstable and therefore
                                                    control (DMC) algorithm the parameters
it is necessary to stabilize the system by
                                                    of the conventional power system
shifting these eigenvalues to the LHP and far
                                                    stabilizer are found.
off from the imaginary axis. The location of
the desired eigenvalues is calculated by
choosing a damping factor  for the dominant
                                                    3.2. GA based design of PSS
                                                              Minimizing the following error
root.         The real part is -n and the
                                                    criteria the controller generates the
imaginary part is
                                                    parameters of the gain and the phase lead
  n  1   2                                      time constant.
             In this paper Integral of time       Lead-Lag compensator was achieved using
  multiplied absolute value of error (ITAE)       the MATLAB/Simulink environment.
  [4] will be minimized through the
  application of a genetic algorithm, as will     3.3. GLS Algorithm
  presently be elucidated. The GA works on        Step 1: set the generation counter k=0 and
  a coding of the parameters (K, T) to be         generate randomly n initial solutions, X0 =
  optimized rather than the parameters            {xi, i=1……….n}. The ith initial solution xi
  themselves. In this study Gray coding was       can be written as xi =[p1 p2…pj…pm],
  used where each parameter was represented       where the jth optimized parameter pj is
  by 16 bits and a single individual or           generated by randomly selecting a value
  chromosome        was      generated      by    with uniform probability over its search
  concatenating the coded parameter strings.      space [pimin ,pjmax]. These initial solutions
  In contrast to traditional stochastic search    constitute the parent population at the
  techniques the GA requires a population of      initial generation x0. Each individual of x0
  initial approximations to the solution. Here    is evaluated using objective function J. set
  30 randomly selected individuals were used      x=x0;
  to initialize the algorithm. The GA then        Step 2: optimize locally each individual in
  proceeds as follows:                            x. replace each individual in x by its locally
                                                  optimized version. Update the objective
   3.2.1. Fitness determination                   function values accordingly.
             The first step of the GA             Step 3: search for the optimum value of the
   procedure is to evaluate each of the           objective function, Jmin. set the solution
   chromosomes and subsequently grade             associated with J min as the best solution,
   them. Each individual was evaluated by         xbest with an objective function of Jbest.
   decoding the string to obtain the Lead-lag     Step 4: check the stopping criteria. If one
   compensator parameters which were then         of them is satisfied then stop, else set
   applied in a Simulink representation of the    k=k+1 and go to step 5.
   closed-loop system.                            Step 5: set the population counter i=0;
     1. The five fittest individuals were         Step 6: draw randomly, with uniform
automatically selected while the remainders       probability, two solutions x1 and x2 from x.
were selected probabilistically, according to     apply the genetic crossover and mutation
their fitness. This is an elitist strategy that   operators obtaining x3;
ensures that the next generation's best will      Step 7 : optimize locally the solution x3 and
never degenerate and hence guarantees the         obtain x3;
asymptotic convergence of the GA.                 Step 8 : check if x3 is better than the worst
2. Using the individuals selected above the       solution in x and different from all
next population is generated through a            solutions in x then replace the worst
process of single-point cross-over and            solution in x by x3 and the value of
mutation. Mutation was applied with a very        objective by that of x3;
low probability of 0.001 per bit.                 Step 9: if i=n go to step 3, else set i=i+1
Reproduction through the use of crossover         and go back to step 6;
and mutation ensures against total loss of any    To demonstrate the effectiveness and
genes in the population by its ability to         robustness of the proposed GLSPSS over a
introduce any gene which may not have             wide range of loading conditions are
existed initially, or, may subsequently have      verified.
been lost.
3. This sequence was repeated until the
algorithm was deemed to have converged (50
iterations). As was indicated previously the
simulation and evaluation of the GA tuned
4. Comparision of various                        Table 1
design techniques
           The linearized incremental state
                                                 PSS parameters of various pss for
space model for a single machine system          single machine system.
with its voltage regulator with four state
variables has been developed. The single               PSS type            PSS parameters
machine system without PSS is found
unstable with roots in RHP. The system            CONVENTIONAL               K = 7.6921
dynamic response without PSS is simulated         LEAD-LAG PSS               T =0.2287
using Simulink for 0.05 p.u disturbance in         GA BASED PSS              K =26.5887
mechanical torque. MATLAB coding is                                           T=0.2186
used for conventional PSS, Genetic PSS,
                                                   GLS BASED PSS             K=35.1469
and Genetic local search (GLS) PSS design
                                                                              T=0.2078
techniques. The final values of gain (K),
and phase lead time constant (T) obtained
from all the techniques are given to the       4.1. Simulation results
simulink block. The dynamic response           1. Performance of fixed-gain CPSS is better
curves for the variables Δω, Δδ and ΔVt are    for particular operating conditions. It may
taken from the simulink. The system            not yield satisfactory results when there is a
responses curves of the conventional PSS       drastic change in the operating point.
(CPSS), GA based PSS as well as GLS            2. Dynamic response shows that the GA
based PSS are compared.                        based PSS has optimum response and the
Shaft speed deviation is taken as the input    response is smooth and it has less over
to the all the Power system stabilizers. So    shoot and settling as compared to
the PSS is also called as delta-omega PSS.     conventional PSS.
The system dynamic response with PSS is        3. As compared to the conventional PSS
simulated using these Simulink diagrams        & GA based PSS the proposed genetic
for 0.05 p.u step change in mechanical         local search (GLS) based design of PSS
torque ΔTm. The dynamic response curves        gives the optimum response and the
for the variables change in speed deviation    response is smooth and it has reduced
(Δω), change in rotor angle deviation (Δδ)     settling time.
and change in terminal voltage deviation       4. The time multiplied absolute value of
(ΔVt) of the single machine system with        the error (ITAE) performance index is
PSS are plotted for three different types of   considered. The simulation results show the
power system stabilizers (PSS) are shown       proposed Genetic local search based PSS
in Figs. 3– 5. It is observed that the         can work effectively and robustly over a
oscillations in the system output variables    wide range of loading conditions over the
with PSS are well suppressed. The Table I      conventional and GA based design of PSS.
shows various types of PSS and its             5. The response curves shows that GLS
parameters after tuned by conventional,        PSS has less over shoot and settling time as
genetic and genetic local search technique.    compared to the GA PSS and the
                                               traditional Lead-lag PSS.
Figure.3 Δ Vs time for normal load condition of the SMIB system




       Fig.4 Δδ Vs time for normal load condition of the SMIB system
                   Fig .5 ΔVt Vs time for normal load condition of the SMIB system
                                                  and robustly over wide range of loading
Conclusion                                        conditions and system configurations.
           In this study, a genetic local
search algorithm is proposed to the PSS
design problem. The proposed design             Appendix
approach hybridizes Genetic Algorithm           Derivation of k-constants
with a local search to combine their                        All the variables with subscript 0
different strengths and overcome their          are values of variables evaluated at their pre-
drawbacks (i.e.) GA explores the search         disturbance steady-state operating point from
space (spectrum of optimum location             the known values of P0 , Q0 and Vt0.
points either maxima or minima) before it                              P0Vto
gives data to the local search technique.       iq 0 =                                         (14)
Then local search technique is the best                    ( P0 x q ) 2  (Vt 2  Q0 x q ) 2
                                                                              0
technique than GA to find the optima
(either maxima or minima). The potential        vd 0 = iq 0 xq                                  (15)
of the proposed design approach has been
demonstrated by comparing the response          vqo = Vt 2  vt20                              (16)
                                                         0
curves of various power system stabilizer
(PSS) design techniques.
Optimization results show that the                       Q0  xqiq 0
                                                                 2

proposed approach solution quality is           id 0 =                                         (17)
                                                            vq 0
independent of the initialization step. Eigen
value analysis reveals the effectiveness and
robustness of the proposed GLSPSS to
                                                Eq 0 = vq0  id 0 xq                           (18)
damp out local mode of oscillations. In
addition, the simulation results show that      E0 = (vd 0  xeiq 0 ) 2  (vq 0  xe id 0 ) 2 (19)
the proposed GLSPSS can work effectively
                  (vd 0  xeiq 0 )                          K4      Demagnetising effect of a change in
 0 = tan 
             1
                                                    (20)    rotor angle
                  (vq 0  xeid 0 )                          K5      Change in Vt with change in rotor
       xq  xd
             '
                                       Eq 0 E0 cos 0       angle for constant Eq’
K1 =               iq 0 E0 sin  0                         K6 Change in Vt with change in Eq’
       xe  x '
              d                           xe  xq           constant rotor angle
                                                    (21)
       E0 sin  0                                          References
K2 =                                                (22)
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     xd  xe
      '
                                                           power       system      stabilizers‖,    IEEE
K3 =                                                (23)
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             '
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       xe  x d
              '
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                                    '
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       xe  xd Vt 0
             '
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