During emergency conditions, one of the main operator`s tasks is to keep
as many customers on-line as possible. For these various, more or less
traditionally tools, such as Generation Reallocation (GR), modification
of exchange schedules with neighboring systems, line switching and finally,
Load Shedding (LS) are available. With minimum operator activity, constraint
violations should be removed and still maximum amount of load served.
During normal operation, the focus is on safe and economical operation
of the power system. This is achieved to typically minimize transmission
losses or production cost. In an emergency situation, safe operation combined
with minimal LSGR is the main objective problem. To do this, power plant
productions are minimally disturbed and minimum amount of load has to
be dropped. Therefore, it can be concluded that the LSGR problems will
carry with itself an optimization process that its objective function
is made up of load shedding and generation reallocation values and constraints
of the above optimization are restriction of system variables.
To solve the LSGR problem, various procedures have been presented. In
most cases, the problem of load shedding have been discussed by Aik (2006),
Hooshmand et al. (1998), Halevi and Kottick (1993), Xu and Girgis
(2001) and since the amount of power production in power plants with due
to continual varying of system frequency, so ignoring generation reallocation
in power plants makes the above mentioned optimization farther from real
conditions over the system. Also in many presented methods (Halevi and
Kottick, 1993; Yao et al., 2005; Xu and Girgis, 2001; Tso et
al., 1997), frequency deviations as an essential variable in optimization
process have not been discussed. Moreover, due to the depending of the
electrical loads of the system on voltage and frequency variations, electrical
load models have not been considered (Halevi and Kottick, 1993; Yao et
al., 2005; Xu and Girgis, 2001; Hsu et al., 2005; Thalassinakis
et al., 2006; Ding et al., 2006; Tso et al., 1997)
and the loads have been discussed as fixed amounts which are far from
the real facts.
Other important problems, which should be considered in load shedding,
are its omission from system busses in periodical order. In other words,
the amount of load shedding can`t be considered as a continuous variable,
because by outage of the feeders in every bus, a specific percent of your
option load is omitted. Therefore, the load variations make the optimization
of the problem not to act as a real case (Hooshmand et al., 1998;
Palaniswamy and Sharma, 1985; Halevi and Kottick, 1993; Xu and Girgis,
2001; Luan et al., 2002; Hsu et al., 2005).
In the published literature on the topic of LSGR problem, different techniques
such as linear optimization (Hooshmand et al., 1998; Halevi and
Kottick, 1993), nonlinear optimization (Aik, 2006; Verzija, 2006), employing
frequency relays (You et al., 2003), dynamic programming (Verzija,
2006; Ding et al., 2006) and intelligent systems have been used.
Also with due to lack of uncertainty in the constraints of power system,
employing the optimization with fuzzy technique in the LSGR problem has
been noticed (Hooshmand et al., 1998; Tso et al., 1997).
In this method, the load shedding values have been considered as continuous
variables. In some studies also load shedding problem have been solved
by using genetic algorithm (Yao et al., 2005; Luan et al.,
2002). Also in Hsu et al. (2005), Purnomo et al. (2002)
and Thalassinakis et al. (2006) application of neural network technique
in load shedding problem has been presented. In addition, in Ding et
al. (2006) and Tso et al. (1997) the load shedding problem
by using a set of dominating rules over power systems is formulated and
then load shedding is done intelligently. But using this technique doesn`t
lead to ideal optimization.
In this study, a new method based on FPSO algorithm for solving the LSGR
optimization problem, together with electric loads and frequency deviation
of the system and other essential constraints is presented. In this method,
the aim is to consider the real ruling conditions over power system variables
||Considering load shedding and generation reallocation
like discrete variables with due to existing and variable steps at
that variation in constructing the alphabet set of FPSO algorithm
||Considering system frequency variations in changes of power plant
productions and load flow equations;
||Considering the real model of electric loads dependency on voltage
||Finally, modeling of power system constraints as well as operational
limitations is considered as good as possible
THE CRISP LSGR FORMULATION
As, it was explained earlier, the LSGR problem is a constrained optimization
problem. The objective function is made up of load shedding and generation
reallocation values. The concerned constraints consist of: load flow limits,
line flow limits, voltage magnitude, maximum and minimum of loads and
generation active and reactive powers, the limit of frequency deviation
and tap-changer of transformers.
THE OBJECTIVE FUNCTION
The LSGR problem can be formulated as an optimization problem, which
attempts to minimize load shedding and generation deviations from the
nominal state. These objectives are transformed into the minimization
of a scalar penalty function J that is the sum of two terms. The first
term penalizes load shedding and the second penalizes deviations in the
generation schedule. More precisely, J is defined by:
where, li and gi are nonlinear penalty functions,
which will take a unique minimum value at the nominal state. One of the
most acceptable objective functions is a quadratic function. Based on
Eq. 1, the quadratic objective function is defined
where, PG (PL) and QG (QL) represent active and reactive powers of a
generator (a load) and represents deviation from nominal value. In appendix
A, it is discussed how the coefficients ai , bi ,
ci and di in the objective function of the above
equation may be selected. If the above mentioned optimization problem
acts in linear form, the best method for linearizing that function is
employing the four piecewise linear portions that causes the least error
in the problem (Hooshmand et al., 1998).
THE PROBLEM CONSTRAINTS
To minimize the linearized objective function of Eq. 2,
the following constraints may be considered:
||Generation constraints: Generators active power
is adjusted by the static response of the governor expressed by:
where, PRi/PRi denotes the governor response
characteristics. Then with above equation, the active and reactive power
generation is considered to lie within its minimum and maximum limits:
||Load constraints: The majority of a power system
loads contain specifications which depend on frequency and voltage
system. So for doing load shedding optimization and precise, it is
appropriate to consider a real model for loads that can be considered
as follows (Hooshmand et al., 1998; Palaniswamy and Sharma,
1985; Purnomo et al., 2002):
In this model, every load can be considered as a combination of load with
fixed power, fixed current of load, fixed impedance and load based on frequency.
As a result, with this load models, power demands should be limited so that,
for instance, even during emergency conditions, a minimum amount of load
is served for a critical bus. Then, the load constraints can be expressed
||Bus voltage magnitude constraints: This constraint
may have different values for a load bus or a generator bus. The range
is, however, limited as follows:
||Frequency limit: Due to the importance of system
frequency variation in controlling emergency conditions, we should
||Transformer tap constraint: A transformer tap
may be effectively employed in the optimization process as a control
variable. Then, by considering the upper and lower limits of tap-changer
constraints, the constraint can be considered as:
||Line flow angle stability constraints: Among
main constraints over transmission lines is angle stability that is
that i and j are the number of send and end buses of transmission lines.
||Load flow constraints: This is set forth as an
equal constraint in the problem. By considering the emergency condition,
presentation of frequency characteristics in load flow equation is
necessary. The following equation should be satisfied in order to
obtain the balance of active and reactive power at each node i:
THE LSGR MODEL
If the problem is considered in a nonlinear form, Eq.
2 as an objective function and Eq. 3-12 are considered
as the constraints of the problem. Presently, if the problem is solved
in a linear method, the objective function and the above constraints should
be linearized. This is accomplished by conventional linearization techniques.
The linearized equations are expressed in terms of the variable increments
ΔPGi, ΔQGi, ΔPLi, ΔQLi,
Δvi, Δδi, Δti, Δf,
where for instance, ΔPGi = PGi and ο
denoted initial value of variables. Thus, the derived Linear Programming
(LP) optimization problem can be expressed as:
Remark: It should be noted that for the deviation of the above
LP problem, the line flow stability Eq. 10 has been
linearized based on the absolute linearization. Also, the system model (load
flow equations) described by Eq. 11-12 has been linearized
by truncation of Taylor expansion. In the last constraints of Eq.
13, J1 to J8 are sub-matrices of Jacobian matrix
and will be:
STRUCTURE OF FPSO ALGORITHM
Review of PSO technique: Since the PSO method is the base of suggested
FPSO method, this section reviews briefly the PSO method. This algorithm
was suggested in (Gaing, 2004). According to this algorithm, a group of
particles (as the variables of the optimization problem) flies in the
searching space. The velocity of each particle is adjusted according its
own movement experience and its companion`s movement experience. It is
clear that in the searching apace, some of the particles have a better
position in keeping the track for the best solution. As a result, other
particles try to reach to the particles that have better situations. Each
particle is aware of both its position with respect to the neighboring
particles and with respect to whole group. It is assumed that the searching
apace is g-dimensional. To simulate this algorithm the following parameters
||No. of particles in the group
||No. of members in the particle
||The number of iteration (or generation)
||The best solution that a particle of the group has achieved up to
||The best solution that the group of particle (as a whole) has achieved
up to now.
|c1 and c2
||Weighting acceleration constants which represents to what extent
each particle pull to pbest and gbest
||Inertia weight factor adjusts the balance between the global and
local explorations in the searching space
||The velocity of particle j at iteration k in the g-dimensional searching
||The position of particle j at iteration k in the g-dimensional searching
||The best previous recorded position of particle j at iteration k
in the g-dimensional searching space
Then, the velocity and position of particle j at iteration can be obtained
where, Rand () and rand () are two random functions that produce random
umber in the range of 0 to 1. Also, in Eq. 14, the inertia weight factor
w is decreased varied linearly from 0.9 to 0.4. In general w can be determined
where, kmax and k are the maximum and current number of iteration
(or generation), respectively.
One of the difficulties of using this algorithm is choosing c1
ALGORITHM OF FPSO
Since the performance of PSO is nonlinear, the use of linear equations
such as Eq. 16 is not suitable. For example, applying
Eq. 16 leads to this issue that the algorithm searches
global solution first and then looks for the local solution. Thus this
creates a linear relationship between the local and global searches.
In the suggested FPSO method, the inertia weight factor w, is varied
according to a nonlinear model. To model the nonlinear variations weight
factor w, the fuzzy logic is used. In order to fuzzify the variation of
factor w, two fuzzy inputs are defined which are described below:
||The first input is called current best performance evaluation
(CBFE) and describes the point that has the best performance. In this
research the normalized form of CPBE, denoted by NCPBE, is used. This
input is determined from:
In this equation, CBPEMIN shows the best performance is to
be guessed that the FPSO is achieved it. Also CBPEMAX is the
worst acceptable performance of FPSO.
||The second fuzzy input is the current value of the inertia
weight factor w.
The fuzzy output is the variation of factor w. The membership functions
of the fuzzy inputs and the fuzzy output are depicted in Fig.
1-3. In Fig. 1-3, S, M and L, stand for small, medium
and large, respectively. Also, the fuzzy rules related the fuzzy system
is given in Table 1.
The FPSO Flowchart: Finally, the structure of each particle in
FPSO algorithm for designing the amounts of load shedding and generation
where, n is the number of population. In order to compute the amounts
of LSGR problem using FPSO algorithm, the following steps should be performed.
||Specify the upper and lower limits of each variable.
||Form the initial population at random with desirable chromosome.
||Choose k =1.
||The fuzzy rules related to the fuzzy system
THE LSGR PROBLEM BASED ON THE FPSO ALGORITHM
Here, we are
determined to explain how to use the proposed FPSO algorithm for the LSGR
optimization problem. The flowchart of the process by utilizing FPSO has
been shown in Fig. 4. In this method, first system data
exit should enter which include specifications of generators, loads, buses,
lines and other constant parameters (stage 1). Also in this stage, the
amounts of load shedding steps for all loads and steps of production variation
of power plants should be specified. These steps of loads are applied
to construct alphabet set of each load variations and generator productions.
If a load is important it can not omitted or reduced, therefore you can
imagine figure zero the load shedding steps in alphabet set. Consequently,
it can be said that the problem is performed with regarding the load importance
and load shedding with real and performable steps in power system.
In the next stage, by doing load flow in normal operation of system,
the initial conditions of optimization process will obtain (stage 2).
Now, emergency case i.e. disturbance in the system, is simulated that
is a case of line tripping, overloading of the lines, generation shedding
(totally or partially), load power increase and etc. (stages 3 and 4).
With doing load flow in the fifth stage, emergency conditions dominating
system variations is specified (6th stage) that whether existing constraints
have been violated or not? If emergency conditions are so, we should think
of doing load shedding and generation reallocation.
For doing so, by employing the FPSO algorithm, first the primary population
is selected at random process considering the elected Alphabet set (stage
7). As mentioned in previous section, genes of each chromosome are the
variation amount in power plant generations and the values of load shedding
of each buses load. Each one of the genes is out of Alphabet set concerned
with it is elected. So for every bus load, the real value of load shedding
steps should be determined which is one of the most important and basic
of this method where load shedding and generation reallocation should
be like a real optimization. In this stage, undesirable chromosomes are
omitted and replaced with desirable ones. By undesirable chromosome, it
is meant that the amounts of generators production of system are equal
or less than the amount of system loads.
||Flowchart for solving the LSGR problem by using FPSO
By forming this process, the desired population becomes convergence and
the best chromosome is determined under the title of real optimization
of LSGR problem for the desired system.
For illustration purposes, the LSGR problem, based on FPSO algorithm
is tested on modified IEEE14-Bus system shown in Fig. 5.
The specifications of this system are provided along with its constraints
in appendix B. For simulation of the above power system, the following
faults (i.e., disturbances) are considered: line outage, generation reduction,
outage of generators, transmission overloads and load increase of buses.
It is necessary to mention that the algorithm is so designed that initially
only generation reallocation (GR) is tried to solve the problem. Provided
that it is unsuccessful, both Load Shedding and Generation Reallocation
(LSGR) will be enabled.
Here, the results on two separate examples are obtained and provided
in Table 2 and 3. It is assumed that
in pre-disturbance condition, the system is operating in its economical
state. Also, the results of various cases of crisp and FPSO modes of the
problem are determined. The details are described in the following parts.
||The simulation results of example I
||The simulation results of example II
Example I: To illustrate the application of the method, it is
assumed that there is 50% (0.2 p.u.) generation loss at bus 11 of the system.
In order to analyze the fault, first the initial load flow has to be done
to determine the initial conditions. Case 1 of Table 2
represents the normal conditions where all system variables are within their
respective limits. Considering the desired generation loss, the results
of load flow are shown in case 2. As may be seen, it is regarded that the
system frequency is reduced to 49.28 Hz. In other words, the power system
has turned on to an emergency state. Now, if we do the crisp optimization
method, we see that by employing the generation reallocation procedure in
power plants, the dominating emergency conditions will remove, that the
LF results are shown in the third case in Table 2. In
this method, the amount of objective function will equal 5.192.
Now, if we employ optimization method with the proposed FPSO algorithm
(case 4), it will be noticed that by employing generation reallocation
in production of the power plants, the emergency conditions have been
eliminated. But, by this method, the amount of objective function (the
same amount of fitness function of desired chromosome) equals 1.762. These
results have been shown in the fourth case. Since, the value of objective
function in the generation reallocation problem indicates the deviations
of the generators production from the nominal state, as a result, the
reduction of this value based on the FPSO method represents a more acceptable
solution compared with the crisp solution.
The same disturbance is reapplied, but now with the load of bus 8 considered
to be dependent on voltage and frequency with the following coefficients
(Palaniswamy and Sharma, 1985; Purnomo et al., 2002):
For all other loads, ppi = qpi = 1.0 and other
parameters are zero. The results of generation reallocation solution,
with crisp and FPSO methods are shown by cases 5 and 6 of Table
2, respectively. In the crisp method, the amount of objective function
is based on Eq. 2 equals 27.925 and while the amount
of fitness function by employing FPSO algorithm reduces 23.41, that shows
more desirable generation scheduling in the generators. Also, in the crisp
optimization method, the system frequency equals 49.929 Hz. Meanwhile
in the proposed method, the frequency system has increased to 49.978 Hz.
With due regard to the many conditions of the case and the practical restrictions
in power systems, application of FPSO technique has had a lot of influence
in finding a real optimal answer applicable in determining the amount
of generation scheduling.
Example II: One of the cases of bringing emergency conditions
up is a sudden overload in power system. In this example, it is assumed
that in bus 10 of Fig. 5, its associated active power
is increased by 1.0 pu (with power factor = 0.82 lag).
||Single line diagram of the modified IEEE 14-Bus system
The normal pre-disturbance results are shown by case 1 of Table
3. Case 2 represents the severe emergency conditions after the disturbance.
As may be seen, the system frequency, the phase angle difference of lines
2 and 3 and voltage magnitude of busses 2, 6, 8, 9, 10 have crossed their
permissible limits. The frequency drop is 0.887 Hz (more than maximum
permissible limit of 0.2 Hz). Thus, the system is in the emergency state.
To eliminate this problem, the crisp and FPSO modes of LSGR problem will
be adopted, where the results are shown by cases 3 and 4 of Table
3, respectively. Also, comparison of cases 3 and 4 indicates that
the value of objective function of the crisp solution is 163.163, but
this amount with FPSO method is reduced to 141.948. These results indicate
that the total deviations in generation schedule and load curtailment
values for the FPSO method are less than the crisp method. Also the amount
of load shedding in the proposed method is less than crisp optimization
method that the problem will cause reducing the dissatisfaction of consumers
in the proposed procedure.
Now, the same disturbance is reapplied, but with load of bus 13 considered
to be dependent on voltage and frequency with the same coefficients as
in Example I. The results of load flows in the cases 5 and 6 of Table
3 confirm that in crisp optimization method, emergency conditions
have been removed by employing load shedding and generation reallocation.
Whereas, if the proposed method is used, it is needless to load shedding
and only by employing generation reallocation, the system will restore
to the normal state. Presently, the amount of objective function in crisp
optimization method is at the amount of 160.057. That will reduce in the
proposed method of the amount of objective function (it is the amount
of fitness function) to 44.583. The reason of abundant decrease in the
amount of objective function in the proposed algorithm will be the nonexistence
of load shedding in the system. These results show that the proposed algorithm
solution for the LSGR problem is more appropriate than the crisp method.
A new method for the optimal LSGR problem based on the FPSO algorithm
in power systems was presented in this study. With due regard to the many
conditions of the case, the practical restrictions in load curtailment
of buses and generation schedule of power plants, application of FPSO
has had a lot of influences in finding a real optimal answer which is
applicable in determining the amount of load shedding and generation reallocation.
In this paper, an attempt has been made to consider frequency variation,
model of electric loads depending on voltage and frequency and other constraints
in optimization problem, system model may be more completely and precisely
In power system case studies, the simulation results explain that FPSO
solution method for the LSGR problem is more flexible than the crisp solution
method. Also, the fitness values of the proposed solution are less than
the respective value (value of objective function) of the crisp solution.
Additionally, the load curtailment values in FPSO case are less than the
corresponding crisp case. Moreover, the proposed approach to the solution
of the LSGR problem accommodates more realistic models to characterize
the behavior of practical power system operations.
The author wish to thank the Department of Research and Technology in
University of Isfahan for supporting this work through the research project
LIST OF SYMBOLS
||Active and reactive power generation
||Load active and reactive powers
||Active and reactive powers injections
||Setting of active and reactive power of a generator
||Setting of active and reactive power of a load
||Rated output and rated regulation of a generator
||Rated voltage and angle of a generator
||Load base voltage value
||Tap value of a transformer
||Fitness value and its normalized
||No. of system buses
||No. of generators
||An element of the admittance matrix Y
||Coefficients of constant impedance load
||Coefficients of constant current load
||Coefficients of constant power load
||Coefficients of frequency dependent part of the load
|ai to di
||Coefficients of the objective function
||Maximum phase angle difference between buses I and j
APPENDIX A. SELECTION OF OBJECTIVE FUNCTION COEFFICIENTS
In the above mentioned problem formulation, the aim is to minimize the
objective function J, so that while the generators operating conditions
minimally disturb from their respective economical operating points, the
least amount of load would be curtailed. Provided that, if the production
cost function of plant is given by:a good practice may be to set:
On the other hand, at rated operating condition, we approximately, have:
where, Sgi is the apparent power. For small perturbation in
Pgi and QGi we have:
In other words, provided the machine is working at rated conditions,
di may be selected as:
The machine is not always working at its rated condition. Hence:
The relative importance of generation reallocation with respect to load
shedding can be considered by ai. In fact, the appreciable
amount of generation reallocation implies that higher operating costs
are tolerable. On the other hand, load shedding produces customer`s dissatisfaction
and surplus unsold energy. Therefore, in an open market environment, generation
reallocation might be worthwhile to a certain extent. Thus, ai
is selected to be:
while, according to a utility experience, other values may also be selected.
With the assumption of fixed load power factors, we have:
||Bus data (initial values before LF) and coefficients
ai`s and bi`s
||Slack and PV bus data and coefficients ci`s
As a result:
APPENDIX B. THE SPECIFICATION OF MODIFIED IEEE 14-BUS SYSTEM
The modified IEEE 14-Bus system Fig. 5 is considered
consisting of five generators, three tap-changer transformers and 20 transmission
lines. The lines, transformers and buses data are given in (You et
al., 2003). The buses data, the minimum (min.) and maximum (max.)
of voltage magnitudes and ai, bi coefficients of
objective function (in conjunction with generators) are shown in Table
B1. The max. and min. of active and reactive generator powers and
ci, di coefficients of objective function (in conjunction
with system loads) are illustrated in Table B 2. The
min. and max. values of tap-changers are 0.9 and 1.1, respectively. Also,
the maximum permissible limit of system frequency has been violated from
nominal 50 Hz to the amount of 0.2 Hz. Finally, it should be mentioned
that for all load buses, the steps of load shedding to the amount of 0.01
per unit are regarded which are used in forming the Alphabet set of each
load of buses.