The power transfer in an integrated power system is constrained by transient
stability, voltage stability and small signal stability. These constraints limit
a full utilization of available transmission corridors. Flexible AC Transmission
System (FACTS) is the technology that provides the needed corrections of the
transmission functionality in order to fully utilize the existing transmission
facilities and hence, minimizing the gap between the stability limit and thermal
limit (Abdel-Magid et al., 1999).
Recently, there has been a surge of interest in the development and use of
FACTS controllers in power transmission systems (Abido and
Abdel-Magid, 2003; Abido, 2002; Al-Baiyat,
2005; Ali and Abd-Elazim, 2011; Ali,
2009a). These controllers utilize power electronics devices to provide more
flexibility to AC power systems. The most popular type of FACTS devices in terms
of application is the SVC. This device is well known to improve power system
properties such as steady state stability limits, voltage regulation and var
compensation, dynamic over voltage and under voltage control and damp power
system oscillations. The SVC is an electronic generator that dynamically controls
the flow of power through a variable reactive admittance to the transmission
network. In last few years, many researchers have posed techniques for designing
SVC to enhance the damping of electromechanical oscillations of power systems
and improve power systems stability. Ali (2009b) uses
a robust control theory in designing SVC controller to damp out power system
swing modes. An Adaptive Network based Fuzzy Inference System (ANFIS) for SVC
is presented by Anderson and Fouad (1977) to improve the
damping of power systems.
Multi input, single output fuzzy neural network is developed for voltage stability
evaluation of the power systems with SVC (Baskaran and Palanisamy,
2006). Chang and Xu (2007) proposes a method of
determining the location of a SVC to improve the stability of power system.
Ellithy and Al-Naamany (2000) presents a systematic
approach for designing SVC controller, based on wide area signals to improve
the damping of power system oscillations. Genetic Algorithm (GA) optimization
technique is employed for the simultaneous tuning of a PSS and a SVC based controller
(Fogel, 1995). A state estimation problem of power systems
incorporating various FACTS devices is addressed (Haque,
2007). A novel hybrid method for simulation of power systems equipped with
SVC is suggested by Kim et al. (2007).
The design of SVC with delayed input signal using a state space model based
on Pade approximation method is presented by Kodsi et
al. (2006). A comparison of Particle Swarm Optimization (PSO) and GA
optimization techniques for SVC controller design is presented by Kundur
(1994). A new optimization algorithm known as Bacterial Foraging Optimization
Algorithm (BFOA) for designing SVC to damp power system electromechanical oscillations
is introduced by Lee and Sun (2002).
Recently, global optimization technique like GA has attracted the attention
in the field of controller parameter optimization by Modi
et al. (2008). Unlike other techniques, GA is a population based
search algorithm which works with a population of strings that represent different
solutions. Therefore, GA has implicit parallelism that enhances, its search
capability and the optima can be located swiftly when applied to complex optimization
Unfortunately, recent research has identified some deficiencies in GA performance
(Panda et al., 2009). This degradation in efficiency
is apparent in applications with highly epistatic objective functions (i.e.,
where parameters being optimized are highly correlated). Also, the premature
convergence of GA degrades its performance and reduces its search capability.
BFOA is proposed as a solution to the above mentioned problems and drawbacks
Moreover, BFOA due to its unique dispersal and elimination technique can find favorable regions when the population involved is small. These unique features of the algorithms overcome the premature convergence problem and enhance the search capability. Hence, it is suitable optimization tool for power system controllers. This study proposes a new optimization algorithm known as BFOA for optimal designing of PI controller for SVC in multimachine power system to damp power system oscillations.
BFOA is used for tuning the SVC controller parameters. The design problem of the proposed controller is formulated as an optimization problem and BFOA is employed to search for optimal controller parameters.
By minimizing the time domain objective function in which the deviations in the local and inter area speed mode are involved; stability performance of the system is improved. Simulations results assure, the effectiveness of the proposed controller in providing good damping characteristic to system oscillations over a wide range of loading conditions and system parameters. Also, these results validate the superiority of the proposed method in tuning controller compared with GA and conventional controller based on Ziegler-Nichols method.
BACTERIA FORAGING OPTIMIZATION: A BRIEF OVERVIEW
The survival of species in any natural evolutionary process depends upon their fitness criteria which relies upon their food searching and motile behavior. The law of evolution supports those species who have better food searching ability and either eliminates or reshapes those with poor search ability.
The genes of those species who are stronger gets propagated in the evolution
chain since, they posses ability to reproduce even better species in future
generations. So, a clear understanding and modeling of foraging behavior in
any of the evolutionary species, leads to its application in any nonlinear system
optimization algorithm. The foraging strategy of Escherichia coli bacteria
present in human intestine can be explained by four processes, namely chemotaxis,
swarming, reproduction and elimination dispersal (Passino,
2002; Rakpenthai et al., 2009).
Chemotaxis: The characteristics of movement of bacteria in search of food can be defined in two ways, i.e., swimming and tumbling together knows as chemotaxis. A bacterium is said to be swimming if it moves in a predefined direction and tumbling if moving in an altogether different direction. Mathematically, tumble of any bacterium can be represented by a unit length of random direction _(j) multiplied by step length of that bacterium C(i). In case of swimming, this random length is predefined.
Swarming: For the bacteria to reach at the richest food location (i.e., for the algorithm to converge at the solution point), it is desired that the optimum bacterium till a point of time in the search period should try to attract other bacteria so that together they converge at the desired location (solution point) more rapidly. To achieve this, a penalty function based upon the relative distances of each bacterium from the fittest bacterium till that search duration is added to the original cost function. Finally when all the bacteria have merged into the solution point, this penalty function becomes zero. The effect of swarming is to make the bacteria congregate into groups and move as concentric patterns with high bacterial density.
Reproduction: The original set of bacteria after getting evolved through several chemotactic stages reaches the reproduction stage. Here, best set of bacteria (chosen out of all the chemotactic stages) gets divided into two groups. The healthier half replaces with the other half of bacteria which gets eliminated, owing to their poorer foraging abilities. This makes the population of bacteria constant in the evolution process.
Elimination and dispersal: In the evolution process, a sudden unforeseen
event can occur which may drastically alter the smooth process of evolution
and cause the elimination of the set of bacteria and/or disperse them to a new
environment. Most ironically, instead of disturbing the usual chemotactic growth
of the set of bacteria, this unknown event may place a newer set of bacteria
nearer to the food location. From a broad perspective, elimination and dispersal
are parts of the population level long distance motile behavior. In its application
to optimization, it helps in reducing the behavior of stagnation (i.e., being
trapped in a premature solution point or local optima) often seen in such parallel
search algorithms. The detailed mathematical derivations as well as theoretical
aspect of this new concept are presented by Yuan et
al. (2010) and Zhijun et al. (2009).
Power system model: A power system can be modeled by a set of nonlinear differential equations are:
||Vector of the state variables
||Vector of input variables
In this study:
and U is the SVC output signals. Here, δ and ω are the rotor angle
and speed, respectively. Also
and Vf are the internal, the field and excitation voltages, respectively.
In the design of SVC, the linearised incremental models around an equilibrium
point are usually employed. Therefore, the state equation of a power system
with n machines and m SVC can be written as:
where, A is a 5x5n matrix and equals
while B is a 5nxm matrix and equals .
Both A and B are evaluated at a certain operating point. X is a 5nx1 state vector
and U is an mx1 input vector.
Modeling of SVC: The Thyristor Controlled Reactor (TCR) in parallel
with a fixed capacitor bank shown in Fig. 1 is used in this
study to develop the desired model. The system is then shunt connected to the
AC system through a set up transformer to bring the voltages up to the required
It is obvious from Eq. 3 and Fig. 2 if the firing angle α of the thyristors is controlled; SVC is able to control the bus voltage magnitude. Time constant (Tr) and gain (Kr) represent the thyristors firing control system:
The variable effective susceptance of the TCR is given by:
where, XL is the reactance of the fixed inductor of SVC. The effective
diagram of SVC|
An auxiliary stabilizing signal from speed can be imposed on the SVC control
loop. The block diagram of a SVC with auxiliary stabilizing signal is shown
in Fig. 2. The differential equation of a PI controller can
be derived as:
where, KP and KI are the gains of PI controller, TW is the wash out time constant. The value of TW is taken as 20 sec. Hence, now the problem reduces to the tuning of gains Kp and KI only.
System under study and SVC location: Figure 3 shows
the single line diagram of the test system used. Details of system data are
given by Zhijun et al. (2009). The system and
generator loading levels are shown in Table 1. In order to
determine the suitable placement of the SVC in the system, two strategies will
be shown below. The 1st one is based on studying the effect of load percentage
while the 2nd is concerned with the line outage on system voltages. Table
2 and 3 show the effect of load percentage and line outage
on bus voltages of the system. It can be noticed that the voltages are affected
significantly at buses numbered 5 and 6, respectively which are load buses.
The reasons that cause the significant voltage change are the connection of
these buses with the longest lines in the system which has greater resistances
and reactances than the others. Consequently, the choice of buses number 5 or
6 for placing the SVC controller is expected to be the more suitable choice.
Loading conditions for the system (p.u.)
Effect of load percentage on load bus voltages
Effect of line outage on load bus voltages
Because both of them are close to machine number 1 which causes the system
instability due to its unstable mechanical mode. Moreover, bus number 5 is the
worst one and will be considered in this study as the best location for installing
the SVC controller.
Objective function: A performance index can be defined by the Integral of Time multiply Absolute Error (ITAE) of the speed deviations of machines. Accordingly, the objective function J is set to be:
The advantage of this selected performance index is that minimal dynamic plant information is needed. Based on this objective function J optimization problem can be stated as: minimize J subjected to:
This study focuses on optimal tuning of SVC using BFOA algorithm. The aim of
the optimization is to search for the optimum controller parameters setting
that reflect the settling time and overshoots of the system. On the other hand,
the goals are improving the damping characteristics and also obtaining a good
performance under all operating conditions and various loads and finally designing
a low order controller for easy implementation.
BACTERIA FORAGING ALGORITHM
In this study, optimization using BFOA is carried out to find the parameters of PI controller for SVC design problem. The algorithm of the proposed technique involves two steps.
||p is the number of parameters to be optimized
||S is the number of bacteria to be used for searching the total region
||NS is the swimming length after which tumbling of bacteria
will be undertaken in a chemotactic loop
||NC is the number of iteration to be undertaken in a chemotactic
||Nre is the maximum number of reproduction to be undertaken
||Ned is the maximum number of elimination and dispersal events
to be imposed over the bacteria
||Ped is the probability with which the elimination and dispersal
||P (1-p, 1-S, 1) is the location of each bacterium which is specified by
random numbers on [-1, 1]
||The value of C (i) which is assumed to be constant in this case for all
the bacteria to simplify the design strategy
||The values of dattract, ωattract, hattract
Iterative algorithm for optimization step-2 : This section models the
bacterial population chemotaxis, swarming, reproduction, elimination and dispersal
(initially, j = k = l = 0). For the algorithm updating θi automatically
results in updating of P:
||Elimination-dispersal loop: l = l+1
||Reproduction loop: k = k+1
||Chemotaxis loop: j = j+1
For i = 1, 2,
, S, calculate cost function value for each bacterium i as follows.
Compute value of cost function J(i, j, k, l). Let:
Jcc is defined by the following equation:
Let Jlast = Jsw (i, j, k, l) to save this value since
one may find a better cost via a run. End of for loop. For i = 1, 2,
S take the tumbling/swimming decision. Tumble generate a random vector, Δ(i)ε
with each element, Δm(i) = 1, 2,
, p. Move let:
Fixed step size in the direction of tumble for bacterium i is considered. Compute
J(i, j+1, k, l) and Jsw(i, j+1, k, l) = J(i, j+1, k, l)+Jcc
(θi((j+1, k, l)), P(j+1, k, l)). Swim, let m = 0 (counter for
swim length). While m<NS (have not climbed down too long). Let
m = m+1. If Jsw (i, j+1, k, l)<Jlast (if doing better),
let Jlast = Jsw (i, j+1, k, l) and let:
and use this θi(j+1, k, l) to compute the new J(i, j+1, k,
l). Else, let m = NS. This is the end of the while statement. Go
to next bacterium (i+1) if I ≠S. If j<Nc, go to step-3. In
this case, continue chemotaxis since, the life of the bacteria is not over.
Reproduction: For the given k and l and for each i = 1, 2, .., S, let:
be the health of the bacterium i (a measure of how many nutrients, it got over
its life time and how successful, it was at avoiding noxious substance). Sort
bacteria in order of ascending cost Jhealth. The Sr =
S/2 bacteria with highest Jhealth values die and other Sr
bacteria with the best value split. If k<Nre, go to step-2. In
this case, one has not reached the number of specified reproduction steps so,
one starts the next generation in the chemotactic loop.
Elimination-dispersal: For i = 1, 2,
, N with probability Ped,
eliminate and disperse each bacterium and this result in keeping the number
of bacteria in the population constant. To do these if you eliminate a bacterium,
simply disperse one to a random location on the optimization domain. If l<Ned
then go to step-2; otherwise end. The detailed mathematical derivations as well
as theoretical aspect of this new concept are presented by Passino
(2002) and Rakpenthai et al. (2009).
In this study, different comparative cases are examined to show the effectiveness of the proposed BFOA method for optimizing controller parameters. Table 4 shows the mechanical modes of system, minimum damping ratio of system mode, performance index and controller parameters. It is clear that the system with conventional controller is suffered from critical damping due to the small damping ratio of system modes (ζ = 0.0536, 0.0099) for light and normal loading, respectively. Moreover, it is unstable at heavy loading condition because of the negative damping of electromechanical modes and damping ratio (S = + 0.07± 8.00 j, ζ = -0.0093). Also, the maximum damping ratio is related to BFOA. Also, the proposed controller shifts substantially the system mode eigenvalues to the left of the S plane and increases the minimum damping ratio of the system (ζ = 0.1179, 0.1417, 0.058) for light, nominal and heavy loading, respectively.
Hence, compared to the conventional PI controller system, BFOA greatly enhances the system stability and improves the damping characteristics of system modes.
Step response for normal load condition: Figure 4-6
show response of system for a 0.1 step increase in mechanical torque of generator
1. In Fig. 4-6, the response with conventional
PI controller is suffered from high settling time and undesirable oscillations.
Also compared with GA, the proposed method is indeed more efficient in improving
the damping characteristic of power system.
in Δω12 for normal load|
Stability of the system is maintained and power system oscillations are effectively suppressed with the application of the proposed controller.
Step response for light load condition: Figure 7-9
show response of Δω12, Δω13, Δω23
for light load condition due to 0.1 step increase in reference voltage of generator
1. From these Fig. 7-9, it can be seen that
the proposed method outperforms and outlasts GA in damping oscillations effectively
and reducing settling time.
Mechanical modes and min ζ under different loading
conditions and controllers
in Δω23 for normal load|
in Δω13 for normal load|
in Δω12 for normal load|
in Δω13 fo normal load|
in Δω23 for normal load|
Hence, compared to the conventional controller and GA based one, SVC based BFOA greatly enhances the system stability and improves the damping characteristics of power system. Moreover, the system with conventional controller cannot reach steady state till 12 sec.
Step response for heavy load condition: In this case, a 0.1 step increase
in mechanical torque of generator 1 is applied for heavy load condition. The
signals of the closed loop system are shown in Fig. 10-12.
in Δω12 for normal load|
in Δω13 for normal load|
in Δω23 for normal load|
It is clear from Fig. 10-12 that the
power system oscillations are increased and system is unstable for conventional
controller. Also, compared with GA the proposed method has a smaller settling
time and system response is quickly driven back to zero. In addition, the potential
and superiority of the proposed method over the conventional and GA in tuning
the parameters of SVC is demonstrated.
Nonlinear time simulation: To evaluate the effectiveness of the BFOA
based SVC tuned using the proposed objective function, a six cycle three phase
fault disturbance at bus 7 at the end of line 5-7 is considered.
in Δω12 for normal load|
in Δω23 for normal load|
in Δω13 for normal load|
The response of Δω12, Δω13, Δω23,
under severe disturbance at t = 1 sec for normal loading is shown in Fig.
It can be seen that the performance of the proposed SVC optimized by BFOA achieves
good robust and provides superior damping in comparison with GA and conventional
case. Moreover, this controller has a simple architecture and the potentiality
of implementation in real time environment.
This study proposes a new optimization algorithm known as BFOA for optimal designing of PI controller for SVC in multimachine power system to damp power system oscillations. The design problem of the proposed controller is formulated as an optimization problem and BFOA is employed to search for optimal controller parameters. By minimizing the time domain objective function in which the deviations in speed are involved; stability performance of the system is improved. Simulations results assure the effectiveness of the proposed controller in providing good damping characteristic to system oscillations over a wide range of loading conditions and different disturbances. Also, these results validate the superiority of the proposed method in tuning controller compared with GA and conventional one over wide range of operating conditions.