A 3 player iterative game to design power system stabilizer that enhances small signal stability

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International Journal of Electrical Engineering & Technology (IJEET)
Volume 6, Issue 7, Jul-Aug, 2015, pp. 38-47, Article ID: 40220150607004
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___________________________________________________________________________
A 3-PLAYER ITERATIVE GAME TO
DESIGN POWER SYSTEM STABILIZER
THAT ENHANCES SMALL SIGNAL
STABILITY
Dr. V. S. Vakula
Asst. Professor, Dept. of EEE, JNTUK UCEV,
Vizianagaram, India.
S. Vamsi Krishna and G Sandeep
PG Student, Dept. of EEE, JNTUK UCEV,
Vizianagaram, India.
ABSTRACT
This paper presents an approach to damp power system oscillations and to
improve small signal stability. Power System Stabilizer must be capable of
providing adequate stabilizing signals over a wide range of disturbances and
operating conditions. Hence, it is very important to appropriately tune the
parameters of the power system stabilizer. Present work is to analyze the
small signal stability of a multi machine test system. To reduce the complexity
in analysis, optimal locations of proposed PSS are found by obtaining
participation factor of respective modes. The paper proposes a systematic
approach to tune the parameters of PSS by defining the interrelations between
inputs and output, i.e., payoff matrix in the iterative prisoner’s dilemma.
Key words: Small Signal Stability, Participation factors, optimal location,
Power system stabilizer and Iterated Prisoner’s Dilemma,
Cite this Article: Dr. Vakula, V. S., Vamsi Krishna, S. and Sandeep, G. A 3-
Player Iterative Game to Design Power System Stabilizer that Enhances Small
Signal Stability. International Journal of Electrical Engineering &
Technology, 6(7), 2015, pp. 38-47.
http://www.iaeme.com/IJEET/issues.asp?JTypeIJEET&VType=6&IType=7
_____________________________________________________________________
1. INTRODUCTION
Power system is a very complicated and complex network consisting of various time
varying loads and generators interconnected together in real time. As a result any
small disturbance in either load or mechanical input to the generator or any

A 3-Player Iterative Game to Design Power System Stabilizer that Enhances Small Signal
Stability
occurrence of fault will induce oscillations in the power system. If these oscillations
remain undamped, they will lead to severe disturbance and in a short time effects
stability of the power system [1]. This results in generator falling out of step, and
finally a system collapse. Thus, it is desirable to damp these oscillations to retain the
system in stable operating condition. Hence, supplementary controllers known as
power system stabilizers (PSS) are used to damp small signal oscillations (0.1–2 Hz)
and to improve the small signal stability of the power system [2].
The conventional step in implementing the PSS is to obtain the optimal locations
in the power system. This reduces the number of PSS required to attain the Small
Signal Stability of the power system. As per the literature, the participation factor
method [3] is widely used to obtain the optimal locations. This involves analysing the
relations between state variables and modes of oscillations. The machines
predominantly participating in the oscillations can be recognized by calculating the
participation factors. Applying PSS at these machines will damp the oscillations
effectively with minimum number of PSS [4].
The challenge involved with the PSS is to tune its parameters such that it performs
robust over a broad range of operating conditions and disturbances in the system [1,
2]. To analyse the system under small disturbance conditions, a linearized model of a
power system [5, 6] can give a satisfactory result. In literature, there are several
methods proposed to tune the parameters of PSS. Genetic algorithm (GA) is one of
the most popular algorithms, and is used by many researchers to aim for optimal PSS
parameter tuning [7–12]. Non dominated sorting GA is used to tuned PSS parameters
in [12]. The GA has its own limitations like premature convergence, requirement of
high computation capacity to solve complex optimization problems and difficulty in
selecting optimum generic operators. Particle swarm optimization (PSO) is from
swarm intelligent family, was used [13–15] to tune PSS parameters. The authors
developed three PSO algorithms based PSS for an interconnected power system
composed of three stand alone power systems in [15]. The disadvantage of PSO is, it
easily suffers from partial optimism which causes less exact at the regulation of speed
and direction.
Prisoner Dilemma (PD) is a classical game, in which each player has two choices
of cooperate (C) and defect (D). Depending on the actions of itself and other players,
each player gets the payoff as output. When the players participate several times in
the game, then it is referred as Iterated Prisoners Dilemma (IPD). In IPD, each player
tries to maximize his payoff by counter acting for its opponent action. The correlation
between the actions of the players and the payoff is denoted by the payoff matrix [17–
19].
In this paper, a 16 machine 68 bus test system [5] is considered. The PSS are
placed at optimal locations. The response of the test system with PSS tuned by IPD is
evaluated. The results are compared with each other. The performance is also
compared GA-PSS and test system without PSS.
2. PRISONER’S DILEMMA
The Prisoner’s dilemma (PD) is a classical game used in ecosystems, social sciences,
economic sciences, control strategies to obtain solutions for various problems [16–
18]. The interaction between two players in PD is defined as a payoff matrix of player
A as in (1).

Dr. V. S. Vakula, S. Vamsi Krishna and G. Sandeep
A
C D
C R S
P
D T P
 
  
 
(1)
Where, C & D are the two strategies known as Co-operation and Defection, that can
be selected by each player in each round, and T > R > P > S. For example, if two
players (A&B) are playing, the payoff matrix elements can be interpreted as below
[19].
 If A & B chooses to co operate, both of them will get the reward R.
 If A & B chooses to defect, both of them deserve punishment P.
 If A or B defects and B or A cooperates, then the defector obtains temptation T as
payoff, while co-operator gets S.
Each player intends to defect to get more payoff, irrespective of the opponent’s
action (T>R and P>S) but if both chooses cooperate, it results in higher payoff over a
long run, 2R>T+S. Generally, as per literature [21], a rescaled form of T>1, R=1,
P=S=0 is adopted. An ith
player’s payoff is obtained as follows:
i
j
i i j
j
P K AK

  (2)
Where, i denotes the neighbourhood of ith
player including itself, A denotes the
payoff matrix of player (1) and   , transpose operation. If ith
player chooses to
defect,  0 1iK  else  1 0iK  .
2.1. Generalized Prisoner’s dilemma (GPD)
GPD is a multi agent or N-Player PD [17], is one where each of N-individual selects
to act between co-operate or defect. As a result each individual receives the payoff for
their actions with respect to other individuals as mentioned above. But to define the
problem of GPD for obtaining appropriate strategy, it should be noted i) whether
individual interact with the others ii) whether individual drops out of the game iii).
Whether individual refuses participate iv) whether actions of individuals is
simultaneous or sequential v) whether individual is known to others actions vi)
whether the game is a one shot or iterated one.
2.2. Iterated Prisoner’s dilemma (IPD)
The iterated prisoner’s dilemma is a game in which there N-players participate several
times i.e., iterations in the game. The player’s action in the next iteration so as to
continue to game can be determined by different strategies mentioned in [21]. Among
those, Tit For Tat (TFT) is the most popular, and variants of TFT are obtained by
combining it with other strategies such as Always Cooperate and Always Defect. The
designed strategy must possess the conditions of being nice, retaliating, forgiving and
clear to serve effective [21].
Present work deals in defining a 3-player iterated strategy to solve an objective
function (15). To reduce the complexity in solving the problem, the rule to discuss the
payoff is implemented in IPD is: If one’s choice is fixed, then the other two players
will result in a two player PD. This results in the following constraints,

Stability
 CCD > DDD
 CCC > DCD
 CCD > (CDD+DCD)/2
 CCC > (CCD+DCC)/2
3. PROPOSED APPROACH
3.1. Test system and modelling
A 16 Machine 68 Bus system is considered [5, 25] as the test system. The state
equation of the linearized model is,
X A X B U     (3)
where '
, , ,
T
i i q tdiX E E         is state vector, A is the state matrix, B is the input
matrix and U is the control vector. The linearized block diagram of the ith machine
with Fuzzy PD PID controller is shown in Figure 1
3.2. Design of Power System Stabilizer
The transfer function of the two stage PSS employed is obtained as,
  1 3
2 4
(1 ) (1 )
( )
1 (1 ) (1 )i
wi i i
PSS i i
wi i i
sT sT sT
V s K s
sT sT sT

  
  
    (4)
where, Ki is the PSS gain, Twi, washout time constant, T1i, T2i, T3i, T4i are the time
constants of two stage lead lag phase compensators. The constants Twi, T2i and T4i are
pre calculated and Ki, T1i and T3i are optimized. This two stage PSS stabilizes the
system response test system by is analysed by minimizing the function in (9). The
SSS of test system with PSS is analysed with 1% step disturbance in mechanical input
at machine 1.
Figure 1 Block diagram representation of ith
machine with PSS

3.3. Optimal PSS location
To reduce the complexity in analysis and reduce the size of the test system model
with PSS, the optimal locations of PSS are found. Thus the optimal locations for the
PSS can be obtained by finding the participation factors of the modes. The
participation matrix (P) combines the left and right eigen vectors as a measure of
association between the state variables and modes [4]
1 2[ ..... ]nP p p p (5)
With 1 2[ .... ]T
i i i nip p p p (6)
1 1 2 2[ ..... ]T
i i i i i ni inp       (7)
Where ki is the kth
entry of the right eigen vector i and ik is the kth
entry of the left
eigen vector i . And the element ki ki ikp   is termed as the participation factor. The
participation factors are the indicatives of the relative participations of the respective
states in corresponding modes. Implementing this method for the test system, it is
found that, the machines 1, 2, 3, 4, 6, 7, 8, 9, 13, 16 are the effective locations for the
placement of PSS. This improves the SSSwith minimum number of PSS.
3.4. Optimal tuning of PSS parameters: IPD-PSS
The parameters of the PSS placed at optimal locations are tuned with IPD and
compared with that tuned by GA in [5]. The objective function is formulated to
optimize a composite set of two objective functions based on eigen values comprising
of real part of the eigen value and damping factor of the mode, thus improving the
SSSof the test system. This can be achieved by minimizing the multi objective
function which is stated below.
Thus, the optimization problem can be stated as,
minimize ( ); n
f x x 
min max
i i iX X X  (8)
( ) max( ) min( )f x    (9)
Where,  is the real part of the eigen value, and  is the damping factor of the
corresponding eigen value. Xi
min
and Xi
max
are the limits for the parameters to be
optimized. The limits for the parameters are as below,
min
310 50, 0.1 1.5i iK T   
Thus tuned parameters of PSS with IPD K = 16, T1 = 0.1, T3 = 1.1 which are with the
fixed values of Tw = 10, T2 = 0.1, T4 = 0.05 as the other parameters of PSS.
4. RESULTS AND DISCUSSIONS
The test system is linearized around an operating point to analyse the small signal
stability. The optimal locations to place optimal PSS are obtained as mentioned in
section 3. The small signal analysis of the test system with optimal parameters of the

Stability
IPD-PSS is compared that with GA [5], i.e., GA-PSS. The modes of oscillations of
the test system for each case are illustrated in Figures 2 to 5.
Figure 2 Modes of oscillations of test system without PSS
Figure 3 Modes of oscillations of test system with GA-PSS[5]
Figure 4 Modes of oscillations of test system with IPD-PSS
From Figure 2, it is evident that the system is unstable as one of its complex pair
of eigen values 0.2 + j6.6 are located on the right side of the S-plane. PSS is essential
to bring back the system to the stable operating condition. From Figure 3, it can be
observed that the modes are shifted to a stable region as none of the modes left on the
right side of the S plane. The SSS of the test system is enhanced as the most left mode
is shifted from -0.35 + j 11.07 to -4.51 + j17.45 i.e., shifted to much more left in the S
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
-15
-10
-5
0
5
10
15
Real axis
Imaginaryaxis
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
-20
-15
-10
-5
0
5
10
15
20
Real axis
Imaginaryaxis
-40 -35 -30 -25 -20 -15 -10 -5 0
-40
-30
-20
-10
0
10
20
30
40
Real axis
Imaginaryaxis

plane which can be observed from Figures 2 and 3. It can also be observed that the
left most mode is shifted from -4.51 + j17.45 to -35.95 from Figures 3 and 4,
improving the stability of the test system. From the eigen values shown in Table 2, it
can be observed that the IPD-PSS gives the minimum value of the combined objective
function (9) and minimum desirable damping ratio   thus acts effectively to
improve the SSS of the test system. The changes in rotor speed at all machines for 1%
step disturbance at machine 1 with IPD-PSS are illustrated in Figures 6 and 7.
Table 1 Eigen value analysis results
Test Case max   min   Objective Function value
Without implementing PSS 0.2018 -0.0305 0.2324
PSS parameters tuned by GA -1 0.1235 -1.1235
PSS parameters tuned by IPD -1.0218 0.0986 -1.1267
Figure 6 Rotor speed deviation of machines 1 to 8 with IPD-PSS
Figure 7 Rotor speed deviation of machines 9 to 16 IPD-PSS
From Figures 6 to 7, it can be observed that the optimal placement of IPD-PSS
results in quick decay of the oscillations when compared with the performance of the
test system without PSS and GA-PSS [5]. The eigen values obtained for different test
cases are tabulated in Table 2.
0 1 2 3 4 5 6 7 8 9 10
-4
-2
0
2
4
6
8
10
x 10
-6
time in seconds
changeinrotorspeed(pu)
machine 1
machine 2
machine 3
machine 4
machine 5
machine 6
machine 7
machine 8
0 1 2 3 4 5 6 7 8 9 10
-1.5
-1
-0.5
0
0.5
1
x 10
-6
time in seconds
changeinrotorspeed(pu)
machine 9
machine 10
machine 11
machine 12
machine 13
machine 14
machine 15
machine 16

Stability
5. CONCLUSION
The work proposes a new approach to design PSS based on Iterative Prisoner’s
Dilemma. Determining the participation factors to identify optimal locations of
placement of PSS has proved that are number of PSS to be incorporated in the test
system is only at 10 machines rather 16 machines. The effectiveness of proposed
approach is validated on a small signal multi machine test system. The performance of
the proposed IPD-PSS is compared with that of GA-PSS. From the results it is
concluded that the IPD-PSS improves damps oscillations by proving more damping
compared to GA-PSS.
Table 2 Eigen values for different test cases
Without PSS GA-PSS[5] Proposed IPD-PSS
-0.35+ j11.08, -0.27+ j9.22,
-0.25+ j8.72,-0.24+ j8.36,
-0.2+ j7.77,0.21+ j7.53,
-0.19+ j7.39,+ j6.9,+ j6.8,
+ j6.75, + j6.15,0+ j6.47,
0+ j4.62,0.01+ j3 .83,
-0.2+ j6.61,-0.02+ j2.06,
-0.1+ j1.83,-0.09+ j1.73,
-0.11+ j1.61, 0.11+ j1.47,
-0.1+ j0.87,-0.1+ j0.95,
-0.11+ j0.98,-0.11+ j1.41,-
0.1+ j1.04, -0.11+ j1.34, -
0.11+ j1.33,-0.12+ j1.26,
-0.09+ j1.26,-0.1+ j1.13,
-0.1+ j1.14, -0.1+ j1.19,
-4.48+ j19.07, -4.46+ j18.03,
-4.52+ j17.46, -4.41+ j15.81,
-4.41+ j15.03,-4.23+ j14.71,
-4.17+ j13.93,-4.41+ j14.1,
-4.46+ j13.97,-4+ j10.41,
-4.44+ j13.41,-3.91+ j12.74,
-4.23+ j13.19, -4.43+ j12.17,
-4.39+ j12.19,-4.05+ j9.09,
-1.23+ j9.91,-1.22+ j5.22,
-1.01+ j1.18,-1.63+ j2.95,
-1.28+ j1.98,-1.25+ j2.13,
-1.07+ j2.62,-1.44+ j4.33,
-1.12+ j4.75,-1.09+ j4.54,
-1.06+ j4.33,-1.09+ j3.55,
-1.23+ j3.85,-1.04+ j3.91,
-1.03+ j3.79,-1.12+ j3.83,-1.
-35.96, -32.47,-30.34+ j2.06,
-30.64+ j1.07,-11.2+ j30.77,
-2.82+ j28.45,-26.32,-11.13+ j29.07,
-7.81+ j28.59,-23.96,-4.38+ j27.15,
-10.8+ j27.27,-6.26+ j27.37,-23,
-3.45+ j23.6, -6.48+ j25.11,
-9.17+ j25.37,-21.17,-8.88+ j23.95,
-10.45+j22.86,-11.04+j21.03,
-4.5+ j21.28,-10.98+ j20.19,
-11.04+j19.65,-2.09+j8.06,-1.33+ j7.22,
-1.15+ j6.87,-1.61+ j6.64,-1.86+ j6.2,
-1.31+ j6.18, -1.15+ j5.62,-1.15+ j6.09,
-1.82+ j4.44, -1.22+ j6.01, -1.03+ j4.15,
-1.56+ j2.82, -1.62+ j4.78,-1.07+ j1.87,
-1.31+ j2.5, -1.3+ j2.22,-1.
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