State Feedback Design for Multi-machine Power
System Using Genetic Algorithm
Abolfazl Jalilvand, Reza Aghmasheh and Amin Safari
Department of Electrical Engineering, Faculty of Engineering, Zanjan University, Zanjan, Iran
ajalilvand@znu.ac.ir , r.aghmasheh.ir@gmail.com and asafari1650@yahoo.com
Abstract-An optimal design of state feedback, as a power
system stabilizer (PSSs) is presented based on genetic algorithm
(GA) by using eigenvalue-based objective functions. The
practical implementation of PSSs and The relative stability of
low-frequency modes are all included in the constraints. The
locally measured states are fed back at the AVR reference input
of each machine after multiplication by suitable feedback gains.
The proposed method is confirmed by eigenvalue analysis and
simulation results for a multi-machine power system under
different operating conditions.
I. INTRODUCTION
Stability of power systems is one of the most important
aspects in electric system operation. This arises from the fact
that the power system must maintain frequency and voltage
levels, under any disturbance [1]. Since the development of
interconnection of large electric power systems, there have
been spontaneous system oscillations at very low frequencies
in order of 0.2 to 3.0 Hz. Once started, they would continue
for a long period of time.
To enhance system damping, the generators are equipped
with power system stabilizers (PSSs) that provide
supplementary feedback stabilizing signals in the excitation
systems. PSSs augment the power system stability limit and
extend the power-transfer capability by enhancing the system
damping of low-frequency oscillations associated with the
electromechanical modes [2]. Several approaches based on
modern control theory have been applied to the PSS design
problem including adaptive [3], robust [4] variable structure
control [5]. These stabilizers provide better dynamic
performance but they suffer from the major drawback of
requiring model parameter identification, state observation
and feedback gain calculations. The eigenvalue sensitivity
analysis has been used for PSS design under deterministic
system operating conditions. It has been used for the
parameter design of power system damping controllers [6].
A new approach for the optimal decentralized design of
PSSs with output feedback is investigated in [7]. If complete
state feedback control scheme are adopted, the requirements
of estimators and centralized controls may be used for the
unavailable states and control signals. These increase the
hardware cost and reduce the reliability of the control system.
Novel intelligent control design methods such as fuzzy logic
controllers [8-9] and artificial neural network controllers [10]
have being used as PSSs. H
∞
optimization techniques have
been also applied; however the importance and difficulties in
the selection of weighting functions of H
∞
optimization have
been reported. The order of the based stabilizer is as same as
the plant. This causes the complex stabilizers and reduces their
applicability [11].
Recently, heuristic methods are widely used to solve global
optimization problems. Techniques such as genetic algorithms
[12], tabu search algorithm [13], simulated annealing [14],
evolutionary programming [15] and swarm optimization
techniques [16] have been applied earlier to the PSSs design.
In this paper the proposed design process for PSSs with state
feedback schemes is applied to a multi-machine power system.
Local and available states ( δ Δ , ω Δ ,
q
E ′ Δ and
fd
E Δ ) are used
as the inputs of each PSS. The design problem is converted to
an optimization problem and GA is employed to solve it.
Robustness is achieved by considering several operating
conditions. The eigenvalue analysis and the time domain
simulation results under various operation conditions are
given, which show that the proposed objective functions
damps the low frequency oscillation in an efficient manner.
II. PROBLEM STATEMENT
A. Power System Model
A power system can be modeled by a set of nonlinear
differential equations as:
( , ) X f XU =
(1)
In this study, the two-axis model is used for time–domain
simulations [2]. When a power system operates at one
operating point, the system is linearized. The model of this
linearized system can be represented by [10]:
() () ()
() [ () ( )]
s d
X t AX t BU t
A X t BU t U t
= +
= + +
(2)
Where the input vector ) (t U is composed of the control
vector ) (t U
S
and disturbance vector ) (t U
d
, both 1 × m
vectors; ) (t X is an 1 × n state vector; A is an n n × plant
matrix of the open-loop system and B is an m n × input
matrix; n and m are the number of state variables and control
signals, respectively.
The following control input is defined as the state feedback:
() ()
s
U t KX t = (3)
Where K is the feedback gain matrix with appropriate
dimensions. Applying (3) to (2):
() ( ) () ()
d
Xt A BK X t BU t = + +
(4)
c
A A BK = + (5)
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