Load shedding and generation reallocation (LSGR) schemes are important and powerful tools in the present day power systems to maintain system stability. In this study, a new method has been presented on the basis of Fuzzy Particle Swarm Optimization (FPSO) algorithm in order to reduce the LSGR in the form of real optimization. By real optimization in this method, it means using the discrete variables for LSGR problem by real steps of load decreasing in every buses and reallocation steps in power plant productions. Also, by considering the system frequency as an essential variable and using the electrical load model with the essential constraints, the LSGR problem is solved. Referring to the above facts and considering the large number of variables in optimization issue, utilization of the FPSO algorithm is very useful in finding the optimal procedure for LSGR problem. Finally, in order to test the proposed method, it has been implemented on the IEEE 14-Bus system and acceptable results have been obtained.
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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 those are:
|•||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 and frequency|
|•||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 by:
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 as:
|•||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 have:|
|•||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 defined as:|
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 are defined:
|n||=||No. of particles in the group|
|m||=||No. of members in the particle|
|k||=||The number of iteration (or generation)|
|pbest||=||The best solution that a particle of the group has achieved up to now|
|gbest||=||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|
|w||=||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 space|
|=||The position of particle j at iteration k in the g-dimensional searching space.|
|=||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 from:
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 from:
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 and c2.
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:|
|Fig. 1:||The membership function of the normalized CBPE|
|Fig. 2:||The membership function of inertia weight factor w|
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. InFig. 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 reallocation is:
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.
|Step 1:||Specify the upper and lower limits of each variable.|
|Step 2:||Form the initial population at random with desirable chromosome.|
|Step 3:||Choose k =1.|
|Fig. 3:||The membership function of variations of inertia weight factor w|
|Table 1:||The fuzzy rules related to the fuzzy system|
|Step 4:||Generate randomly in the specified limit.|
|Step 5:||Compute the evaluation function f(xj) (which is the objective function shown in Eq. 2 and assign it to Pbestkj, for j = 1,2,...,n.|
|Step 6:||Determine Gbest from Gbest = min (f(pbest j ))|
|Step 7:||Compute from Eq. 14. Note that in the FPSO algorithm, the inertia weight factor w which is needed for Eq. 14, is obtained according to the fuzzy logic discussed in this study.|
|Step 10:||Compute from Eq. 15.|
|Step 11:|| |
Update Pbest j according to the following:
|Step 12:||Update Gbest according to the following:|
|Step 13:||Set k = k+1.|
|Step 14:||Determine Gbest for the xoptimal or continue, as:|
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.
|Fig. 4:||Flowchart for solving the LSGR problem by using FPSO Algorithm|
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.
|Table 2:||The simulation results of example I|
|Table 3:||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).
|Fig. 5:||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 simulated.
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 No. 860819.
LIST OF SYMBOLS
|PG, QG||:||Active and reactive power generation|
|PL, QL||:||Load active and reactive powers|
|P, Q||:||Active and reactive powers injections|
|Pgset, Qgset||:||Setting of active and reactive power of a generator|
|PLset, QGset||:||Setting of active and reactive power of a load|
|PR, R||:||Rated output and rated regulation of a generator|
|Vi, δi||:||Rated voltage and angle of a generator|
|VLB||:||Load base voltage value|
|t||:||Tap value of a transformer|
|fi, gi||:||Fitness value and its normalized|
|NB||:||No. of system buses|
|NG||:||No. of generators|
|Yij, θij||:||An element of the admittance matrix Y|
|pz, qz||:||Coefficients of constant impedance load|
|pc, qc||:||Coefficients of constant current load|
|pp, qp||:||Coefficients of constant power load|
|kp, kq||:||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:
|Table B1:||Bus data (initial values before LF) and coefficients ai`s and bi`s|
|Table B2:||Slack and PV bus data and coefficients ci`s and di`s|
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.
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