Electrical Power and Energy Systems 43 (2012) 304–312
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An enhanced RCGA for a rapid and reliable load flow solution of electrical power systems Hassan Kubba ⇑, Hazlie Mokhlis Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
a r t i c l e
i n f o
Article history: Received 26 January 2011 Received in revised form 11 April 2012 Accepted 20 April 2012 Available online 26 June 2012 Keywords: Genetic algorithms Load flow analysis Load modeling Modeling Sparse matrices Simulation
a b s t r a c t The paper presents a reliable and fast load flow solution by using a real-coded genetic algorithm (RCGA), bus reduction technique and sparsity technique. The proposed load flow solution firstly used reduction technique to eliminate the load buses. Then, the power flow problem is solved for the generator buses only using real-coded GA to calculate the phase angles. Thus, the load flow problem becomes a single objective function, where the voltage magnitudes are specified resulted in reduced computation time for the solution. Once the phase angle has been calculated, the system is restored by calculating the voltages of the load buses in terms of the calculated voltages of the generator buses. A sparsity technique is used to reduce the computation time further as well as the storage requirements. The proposed load flow solution also can efficiently solve the load flow problems for ill-conditioned power systems whereas the conventional RCGA alone fails to solve these systems. The proposed method was demonstrated on 14-bus IEEE, 30-bus IEEE and 300-bus IEEE, and a practical system 362-busbar Iraqi National Grid. The proposed solution has reliable convergence, a highly accurate solution and much less computing time for on-line applications. The method can conveniently be applied for on-line analysis and planning studies of large power systems. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The power flow problem is the determination of voltage magnitudes, voltage phase angles, power flow, power factor, and losses at each busbar and at any point in the power system under existing or contemplated conditions. To find the steady state operating condition of a system is very important and the most frequently carried out study by electrical power utilities for power system on-line operation, planning and control. The mathematical formulation of the electrical power flow problem results in a set of non-linear algebraic equations. The optimization numerical methods such as Newton–Raphson method [1] or the artificial intelligence methods such as genetic algorithm (GA) are applied to solve the power flow problem. The power flow problem has multiple solutions [2]. The numerical methods and some of the artificial intelligence methods suffer from the local minima problem. Also there are many criteria which should be taken into consideration such as the speed of solution, storage requirement and the degree of solution accuracy. With increasing computer speeds, researchers are increasingly applying artificial and computational intelligence techniques in ⇑ Corresponding author. Address: Al-Jadriya, Baghdad University, Engineering College, Electrical Engineering Department, Baghdad, Iraq. E-mail addresses:
[email protected],
[email protected] (H. Kubba),
[email protected],
[email protected] (H. Mokhlis). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.034
power system problems. These techniques offer several advantages over traditional numerical methods such as ability to address nonlinear problem and complex conditions. Among these techniques is that of genetic algorithm. Genetic algorithms (GAs) are efficient stochastic search algorithms that emulate evolution. They have been used successfully to solve a wide range of optimization problems. Because of existence of local minima, these algorithms offer promise in solving large-scale problems. A genetic algorithm imitates Darwin’s evolution process by implementing ‘‘survival of the fittest’’ strategy. Genetic algorithm solves linear and nonlinear problems by exploring all regions of the search space and exponentially exploiting promising areas through selection, crossover, and mutation operations. They have been proven to be an effective and flexible optimization tool that can find optimal or near-optimal solutions [3]. Although GAs are often used to solve the optimal load such as in [4,5], very few researches solving the load flow problem using GAs [6,7]. This is mainly due to the high computation time for a real-time solution. Kubba in [8] had used sparsity technique only to reduce the total computation time as well as storage requirement. Another approach based on genetic algorithms to find multiple load flow solutions is presented by Yin [6]. Advanced constrained (adjusted) genetic algorithm load flow method in [3] is different from the conventional (unadjusted) load flow which is our problem. In constrained load flow, there are two imposed conditions which are
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
305
Nomenclature N EG EL Y Ek ci, di sp Gik, Bik m
number of busbars in the system M-dimensional vector of voltages of generator busbars (N–M) dimensional vector of voltages of load busbars admittance matrix of order N N conjugate of kth busbar voltage real and imaginary parts of ai specified value real and imaginary parts of the admittance Yik number of generator busbars in the system
checked at every iteration (generation) which the reactive power at each generator bus should within Qmin and Qmax as well as the voltage at each load bus should within Vlmin and Vlmax The constrained load flow is usually used for security assessment study. A genetic algorithm with sharing scheme (Newton–Raphson) is proposed and applied to simultaneously find multiple load flow solutions, but the total computation time is still problematic. In this study, a load flow solution combining three main techniques; real-coded genetic algorithm, a method of reduction and restoration of the power system, and sparsity technique is proposed. By combining these techniques, a reliable and fast solution could be obtained in solving load flow problem. The proposed load flow solution also has the capability to solve ill-conditioned power systems. The test on different type of power system networks demonstrates its robustness and practicality for on line analysis. 2. Description of the enhanced real-coded genetic algorithm based load flow solution The real-coded genetic algorithm is very similar to the binary genetic algorithm with the main different that variables are represented in a floating-point rather than a bit of one or zero as in the binary genetic algorithm [10]. We will present the RCGA operators, which are used in this research. 2.1. The variables and cost function of the load flow problem In load flow problem of this research cost function is efficiently used as an objective function. Suppose the chromosome has Mvar variables (a 2M-dimensional minimization problem) given by (a1, a2, . . ., aMv ar ) where M is the number of buses, then the chromosome is written as an array with (1 Mvar) elements so that:
chromosome ¼ ½a1 ; a2 ; a3 ; . . . ; aMv ar
ð1Þ
In power flow problem, the chromosome is written in terms of variables for the voltage magnitude and voltage phase angle of all the buses as follows:
chromosome ¼ ½V 1 ; V 2 ; . . . ; V M ; h1 ; h2 ; . . . ; hM
ð1:1Þ
In this case, the variable values are represented as real numbers. Each chromosome is associated by its cost function, found by evaluating the cost function (f) of the variables (V1, V2, . . ., VM, h1, h2, . . ., hM).
cost ¼ f ðchromosomeÞ ¼ f ða1 ; a2 ; . . . ; aMv ar Þ
ð2Þ
Eqs. (1.1) and (2) along with imposed constraints constitute the problem to be solved. Our primary problem in this research is the continuous functions introduced below. The two cost functions are DPi and DQi: M X DPi ¼ Psp V k ðGik cos hik þ Bik sin hik Þ i Vi k¼1
ð3Þ
IG IL Y11, Y12, ek, fk cal CPU V h
M-dimensional vector of currents of generator busbars (N–M) dimensional vector of currents of load busbars Y21, Y22 sub-matrices of Y of appropriate order inphase and quadrature components of Ek calculated central processing unit voltage magnitude voltage phase angle
where P sp i is the specified active power at bus i; Eq. (3) is for ‘‘PV’’ (generator buses), and ‘‘PQ’’ (load buses), and
DQ i ¼ Q sp i Vi
M X V k ðGik sin hik Bik cos hik Þ
ð4Þ
k¼1
where Q sp i is the specified reactive power at bus i. Eq. (4) is for PQ buses only, where, hik = hi hk, and (DPi) is the mismatch active power at bus (i) and DQi is the mismatch reactive power at bus (i). (Vi, Vk, hi, hk) are the voltage magnitudes and angles at busses (i) and (k) respectively, which are the variables of the two cost functions and M is the number of buses [11,12]. 2.2. Variable encoding, precision, and bounds The difference between binary and real-coded genetic algorithms is now shown [11]. It is not needed to consider how many bits are necessary to represent accurately a value. Instead, (V) and (h) have continuous values that are limited between appropriate bounds for a stable power system which are in our problem, 0.9 6 V 6 1.1 and 20° 6 h 6 20°, for a small-scale power system with light loads, the range of V and h are taken 0.95 6 V 6 1.05 and 5° 6 h 6 5°. Since the genetic algorithm is a search technique, it must be limited to exploring an acceptable region of variable space. Sometimes, this is done by imposing a constraint on the problem or by variables normalization. If there is no experience about the initial search space, and we want to explore a suitably sized variable region before searching on the most promising spaces, thus a genetic algorithm with sharing (may be Newton– Raphson method) to improve choosing the initial search region. 2.3. Initial population The genetic algorithm starts with a group of chromosomes, i.e., the population while each chromosome represents a solution to the problem. A matrix represents the population with each row in the matrix being a (1 Mvar) array (chromosome) of continuous values. Given an initial population of Mind chromosomes while number of chromosomes depends on the power system size and experience, the full matrix of (Mind Mvar) random values is generated since the GA is stochastic in nature. All variables are normalized to have values between 0 and 1. The values of a variable are unnormalized in the cost function. If the range of values is between alo and ahi, then the unnormalized values are given by [11]:
a ¼ ðahi alo Þanorm þ alo
ð5Þ
where ahi is highest number in the variable range, alo is lowest number in the variable range, and bnorm is normalized value of variable. The population size is kept constant throughout the whole solution process.
306
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
2.4. Natural Selection A natural selection is a simple one while we remove the chromosomes with the higher costs. First, the Mind costs and associated chromosomes are sorted from lowest cost to highest cost in an ascending pattern. Then, only the best are selected to continue, while the rest are deleted. The selection rate, Rrate, is the fraction of Mind that survives for the next step of mating and Rrate is selected according to the experience of the analysts, different values of Rrate was tried then, we found Rrate = 0.5 (50%) is the best for our problem. The number of chromosomes that are kept each generation is:
M remain ¼ Rrate Mind
ð6Þ
Natural selection occurs at each generation or iteration of the algorithm. Of the Mind chromosomes, only the top Mremain survive for mating, and the bottom (Mind Mremain) are discarded to make room for the new offspring. Another approach to natural selection is called thresholding (Truncation Selection) which is used also in this research. In this approach, all chromosomes that have a cost lower than some truncation threshold survive. At first, only a few chromosomes may survive. In later generations, however, most of the chromosomes will survive unless the threshold is changed. In this technique the population does not have to be sorted thus, reducing the CPU time for a converged solution [11]. 2.5. Selection
where b = a random number on the interval [0, 1], amn = nth variable in the mother chromosome, adn = nth variable in the father chromosome. The same variable of the second offspring is the complement of the first (i.e. replacing b by 1 b). When b = 0.5, the result is an average of the variables of the two parents. Heuristic crossover is a variation where some random number b is chosen on the interval [0, 1] and the variables of the offspring are defined by:
anew ¼ bðamn adn Þ þ adn
Variations on this theme include choosing any number of variables to modify and generating different b for each variable. This method also allows generation of offspring outside of the values of the two parent variables. Sometimes, values are generated outside of the allowed range. If this happens, the offspring is discarded and the algorithm tries another b. In our problem, we want to find a way to closely imitate the advantages of the binary genetic algorithm scheme. We will chose a crossover point c
parent 1 ¼ ½am1 ; am2 ; . . . ; amc ; . . . ; amNv ar ; parent 2 ¼ ½ad1 ; ad2 ; . . . ; adc ; . . . ; adNv ar ; where (m) and (d) subscripts discriminate between the mom and dad parent. Then, the selected variables are combined to form new variables that will appear in the children:
anew1 ¼ amc bðamc adc Þ; Selection is the first GA0 operators, which is applied on the chromosome. In this process, two chromosomes selected from the mating pool of Mremain chromosomes to produce two new offspring. Pairing takes place in the mating population until (Mind Mremain) offspring are born to replace the discarded chromosomes. Two types of selection are used in this research, which are [7,8]. 2.5.1. Rank-weighted roulette wheel This approach uses a uniform random number generator to select chromosomes. The row numbers of the parents are found using MATLAB V.7 as in this research or any other software. The chromosomes are ranked according to their values of the cost function. The probability of the chromosomes to be selected for mating process is found from the rank of the chromosome. 2.5.2. Tournament selection A small subset of chromosomes, two or three are randomly picked from the mating pool and the chromosome with the lowest cost in this subset becomes a parent. The typical value is k = 2 (socalled tournament size). The tournament repeats for every parent needed. Thresholding and tournament selection make a nice pair, because the population never needs to be sorted. 2.6. Recombination (crossover) As for the binary algorithm, two parents are chosen, and the offspring are some combination of these parents. The problem with these point crossover methods RCGA is that no new information is introduced: each continuous value that was randomly initiated in the initial population is spread to the next generation, only in different combinations. To overcome this problem, we should use two individuals (two parents) and combining their variable values to produce a new variable values in the offspring anew, this method is called blending method [11,13]. A single offspring variable value anew comes from a combination of the two corresponding parent variable values:
anew ¼ bamn þ ð1 bÞadn
ð7Þ
ð8Þ
anew2 ¼ adc þ bðamc adc Þ
where b is also a random value between 0 and 1. The final step is to complete the crossover with the rest of the chromosome as in binary genetic algorithm [8,9]:
offspring 1 ¼ ½am1 ; am2 ; . . . ; anew1 ; . . . ; adNv ar ; offspring 2 ¼ ½ad1 ; ad2 ; . . . ; anew2 ; . . . ; amNv ar This blending method does not permit offspring variables outside the bounds set by the parent unless b > 1. 2.7. Mutation Random mutations vary a certain percentage of the genes or alleles in the schedule of chromosomes. In many solutions after selection and recombination operators the solution can converge too quickly into one region of the cost space. However, the load flow problem has many local minima. If nothing is done to solve this tendency to converge quickly, it may end up in a local minimum. This premature convergence can be prevented to explore other area in the search space by mutation. Mutation points are randomly selected from the (Mind Mvar), total number of genes in the population matrix. Increasing the number of mutations increases the algorithm’s freedom to search outside the variable search space. With the crossover and mutation process, there is a high chance that the optimum solution could be lost as there is no guarantee that these operators will preserve the fittest string. To remedy this, elitist models are often used. In an elitist model [10], the best individual in the population is saved before any of these operations take place. After the new population is formed and evaluated, it is examined to see if this best structure has been preserved. If not, the saved copy is reinserted back into the population. The genetic algorithm then continues on as normal. 3. Method of power system reduction and restoration A fast load flow method proposed in [2,14] is adopted in this study. In this method, the load busbars are eliminated, retaining only generator busbars for the iterative process. Once, obtaining
307
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
convergence, the original network is restored and the voltages of the load busbars are calculated. Overall, computation time of load flow can be reduced. The system equations in terms of generator busbars and load busbars can be written as:
IG
¼
IL
Y 11
Y 12
Y 21
Y 22
EG
ð9Þ
EL
If the voltage of the Kth load busbar is initially assumed to be . Then the current in the busbar to the load is
ILk ¼
PLkjQ Lk ELk
ð10Þ
From the second row of Eq. (9), we have 1 EL ¼ Y 1 22 Y 21 EG þ Y 22 IL
ð11Þ 5. Algorithm of the proposed method (enhanced real-coded genetic algorithm)
Substituting Eq. (11) in the first row of Eq. (9), we get 1 IG ¼ Y 11 EG þ Y 12 ðY 1 22 Y 21 EG þ Y 22 I L Þ
ð12Þ The algorithm for the proposed method is as follows:
Thus, the above equation is written as
IG ¼ Y GG EG þ Y GL IL
ð13Þ
where
Y GG ¼ Y 11 Y 12 Y 1 22 Y 21
and Y GL ¼ Y 12 Y 1 22
ð14Þ
From Eq. (12), the ith generator busbar is:
Ii ¼
m X Y ik Ek þ r i ;
for i ¼ 1; 2; . . . ; m:
ð15Þ
k¼1
where m is the number of generator buses and ri is the ith element of the column vector R given by,
R ¼ Y GL IL
ð16Þ
The complex power at the busbar is
Si ¼ Ei Ii ¼ Ei
m X Y ik Ek þ Ei r i
for i ¼ 1; 2; . . . ; m:
ð17Þ
k¼1
The real power injection at the busbar is
Pi ¼ Re Si ¼
m m X X ei ðek Gik fk Bik Þ þ fi ðek Bik þ fk Gik Þ þ Li k¼1
For i ¼ 1; 2; . . . ; m and E ¼ e þ jf
ð18Þ
k¼1
or E ¼ Vejh
where
Li ¼ ei ci þ fi di
matrix is a highly sparse matrix. The implementation of sparsity technique in MATLAB, uses three arrays internally to store sparse matrices with real elements. Let an (m-by-n) sparse matrix with (NNZ) nonzero entries. The first row vector (array) consist all the nonzero elements of the array in real-coded format. The dimension of this row array is equal to (NNZ). The second row vector contains the corresponding integer row indices for the nonzero elements. This array also has a dimension equal to (NNZ). The third array contains integer pointers to the start of each column. This array has a dimension equal to (n). This matrix requires storage for (NNZ) floating-point numbers and (NNZ + n) integers. Let 8 bytes per floating-point number and 4 bytes per integer, the total number of bytes required to store a sparse matrix is: Grand total of bytes = 8 NNZ + 4 (NNZ + n).
ð19Þ
where (Li) can be considered as an equivalent local load at generator busbar i due to elimination of the load busbars. 4. Sparsity techniques Sparse matrix is a matrix that consist a significant number of elements of zero-valued. This feature allows to [15,16]: Store only the elements of nonzero-valued of the matrix, with their indices, to reduce the needed memory. Reduce the computing time for any arithmetic operation by reducing the operation on zero elements.
1. Read the lines data and form the sparse nodal admittance matrix using sparsity technique as follows: Enter three input arrays, the first array contains all the nonzero elements in real values format. The second array contains the corresponding integer row indices for the nonzero elements. The third array contains integer pointers to the start of each column. 2. Read the busbars data, such as the specified active power, specified voltage magnitude of the generator buses, specified active and reactive power of the load buses, slack bus voltage, and initial estimate of the voltage of the load buses, assuming (1.0 p.u.), the per-unit quantities are 100 MVA and 132 kV. 3. Eliminate the load busbars and reduce the network to the size of that of the generators busbars. 4. Compute (IL) using (10) for all load buses, form the column vector (R) given by (16), then form (Li) assuming (ei) equal to the specified values, and (fi) initially is zero. 5. Execute the real-coded genetic algorithm on the generator buses only to find the most recent value of the voltages, implementing all the GA operators such as Selection with Rank-Weighting Roulette Wheel, Tournament selection with truncation threshold, Single-point Crossover with blending method, and Mutation (rate of mutation = 0.2; increasing the number of mutations may increase the algorithm’s freedom to search outside the current region of variable space). We use initial population of 20 chromosomes for 14-bus IEEE system, 200 chromosomes for 30-bus IEEE system, 400 chromosomes for 300-bus IEEE, and 500 chromosomes for Iraqi National Grid (ING) system, taking into consideration the number of variables for each system. At each generation of the GA, we calculate the most recent values of (EL) from (11), (IL) from (10) and (Li), then calculate (Pi) from (18). 6. Convergence test: The mismatch active powers for the generator buses (cost function) are calculated at each GA generation (iteration) according to the following equation: cal DPi ¼ P sp i Pi ;
P cal i
The type of sparsity technique in this research is done by choosing three arrays to store sparse matrices with real elements. The density of a matrix is defined as the number of elements with nonzero valued divided by the total number of matrix elements. In our research the input data representing by the nodal admittance
for i ¼ 1; 2; . . . ; m:
ð20Þ
where is the calculated active power at each generator bus (i) using (18). When the mismatch active powers (cost function) for all generator buses except the slack bus are less than a small standard tolerance value, usually 0.001 p.u. (0.1 MW/MVAr), then the solution has converged.
308
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
7. Restore the system and calculate the load busbar voltages using (11). 8. Print results and end.
Table 2 Power flow solution (14-bus IEEE) system with an accuracy of (0.001), using enhanced RCGA method.
6. Simulation and testing results Four test systems were used to demonstrate the performance of the proposed method, namely: (1) 14-busbar IEEE test system; the line and bus data are presented in [21]. The ‘‘14-bus’’ test system consists of: 1 slack bus, 4 generator buses (PV) and 9 load buses (PQ) with 20 branches. (2) 30-busbar IEEE test system: this consists of 1 slack/swing bus, 5 generator (PV) buses, and 24 load (PQ) buses with 41 branches. The line and bus data are present in [22]. (3) 300-busbar IEEE test system: this consists of 1 slack bus, 62 generators bus, and 237 load buses with 411 branches. The bus and line data are available [23]. (4) The Iraqi National Grid (ING) which contains ‘‘362 busbars’’ consists of: 1 slack bus, 29 generator buses (PV) and 332 load buses (PQ) with 599 branches [17]. Some specifications of ING are present in Appendix A. The load flow solutions using conventional real-coded genetic algorithm programs and the proposed method have been developed by the use of Matlab version7, and tested with a Pentium 4, 3 GHz, Cache 2 M, PC with 2 GB RAM. The simulation and implementations had been made by two categories of tests on well and ill-conditioned power systems.
ð21Þ
i¼1
where fi is the cost function (i), wi is the weighting factor, h is the P number of objective functions, and hi¼1 wi ¼ 1 Implementing this multiple objective optimization approach in a real-coded genetic algorithm only requires modifying the cost
Table 1 Power flow solution for ‘‘14-bus’’ IEEE test system with an accuracy 0.001 p.u using conventional RCGA. Active power mismatch
Reactive power mismatch
Voltage magnitude (p.u)
Voltage angle (°)
No. of generations
1 2 3 4 5 6 7
Slack 0.00032 0.00013 0.00048 0.00089 0.00079 0.00036
Slack PV PV 0.000481 0.000773 PV 0.000060
1.06 1.045 1.01 1.017131 1.025362 1.07 1.063818
0.00 3.2117 14.358 6.1436 8.423 10.102 13.654
– 15 7 21 47 95 107
8 9 10 11 12 13 14
0.00022 0.00018 0.00027 0.00095 0.00041 0.00077 0.00020
PV 0.000682 0.000322 0.000223 0.000535 0.000521 0.000762
1.09 1.0656395 1.0516310 1.0769601 1.0617250 1.0582889 1.0537880
12.712 14.408 9.0031 15.482 15.677 11.028 13.344
193 172 18 90 43 29 47
Bus no.
Total computing time: 7.156 s
Reactive power mismatch
Voltage magnitude (p.u)
1 2 3 4 5 6 7
Slack 0.000373 0.000130 0.000374 0.000890 0.000678 0.000360
Slack PV PV 0.00044 0.00078 PV 0.0006
1.06 1.045 1.01 1.056131 1.06092 1.07 1.04888
0.00 3.2117 3.35826 6.10062 2.7120 6.30252 4.6999
– 6 5 – – 21 –
8 9 10 11 12 13 14
0.000223 0.000109 0.000223 0.000850 0.000407 0.000660 0.000205
PV 0.0005 0.00032 0.00022 0.00054 0.00051 0.00076
1.09 1.058895 1.051690 1.05990 1.042725 1.04889 1.048837
10.412 1.63081 9.03316 4.08283 6.60054 11.0208 3.34469
40 – – – – – –
Table 3 Power flow solution for ‘‘30-bus’’ IEEE test system using enhanced RCGA, power mismatch = 0.001 p.u.*.
Table 1 illustrates the power flow solution for a 14-bus IEEE test system using RCGA with two objective functions, which are the mismatch active and reactive powers at each bus according to its constraints except the slack bus (Eqs. (3) and (4)). The sum of weighted cost multi-objective functions is used, the most straightforward approach to multi-objective optimization is to weight each function and add them together [20]. h X wi fi
No. of generations
Active power mismatch
Total computing time: 0.18 s
6.1. Load flow solutions of well-conditioned power systems
cost ¼
Voltage angle (°)
Bus no.
Bus no.
No. of generations
Voltage mag.
Voltage angle (°)
DP (p.u)
DQ (p.u)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Slack 40 – – 27 – – 22 – – 28 – 67 – – – – – – – – – – – – – – – – –
1.060 1.043 1.022 1.016 1.010 1.012 1.006 1.010 1.029 1.006 1.082 1.031 1.071 1.007 1.008 1.007 1.000 1.000 0.997 1.000 0.997 0.998 0.999 0.994 1.001 0.997 1.007 1.008 1.000 0.998
0.000 0.397 0.156 0.321 4.002 0.648 1.864 1.02 0.269 0.557 0.052 0.772 0.306 0.338 0.414 0.266 0.464 0.309 0.42 0.352 0.552 0.476 0.279 0.425 0.091 0.22 0.167 0.377 0.195 0.641
– 0.0008 0.0007 0.0009 0.0033 0.0011 0.0024 0.0009 0.0006 0.0007 0.0002 0.0007 0.0006 0.0005 0.0006 0.0005 0.0006 0.0004 0.0005 0.0005 0.0006 0.0006 0.0004 0.0004 0.0002 0.0003 0.0004 0.0009 0.0003 0.0007
– PV 0.0009 0.0008 PV 0.0004 0.0003 PV 0.0007 0.0003 PV 0.0007 PV 0.0007 0.0006 0.0007 0.0003 0.0001 0.0 0.0 0.0 0.0 0.0 0.0 0.0001 0.0001 0.0004 0.0005 0.0 0.0001
Total computation time = 2.6 s *
Total computing time using conventional RCGA = 30.66 s
function to fit the form of (21) and does not require any modification to the genetic algorithm. Thus, (21) becomes:
cost ¼ wf1 þ ð1 wÞf2
ð22Þ
where f1 and f2 are the mismatch active and reactive powers respectively, and have the same rank of importance. This approach is adopted in this work for its simplicity, ease of programming and
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
309
Table 4 Power flow solution for ‘‘Iraqi National Grid’’ with an accuracy of 0.001 p.u using enhanced RCGA, only PV buses.** Bus no.
Active power mismatch (p.u)
Reactive power mismatch (p.u)
Voltage magnitude (p.u)
Voltage angle (°)
No. of generations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Slack 0.0005 0.0002134 0.000808 0.0001125 0.0004310 0.001847 0.0006643 0.0002882 0.0001664 0.000391 0.0004558 0.0001362 0.0005891 0.0004176 0.0002106 0.0007820 0.000477 0.0009163 0.0008909 0.0003712 0.0001452 0.0009387 0.0008442 0.0003853 0.0002358 0.000.204 0.0001168 0.0002683 0.000579
Slack PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV PV
1.04 1.048 1.000 1.001 1.001 1.001 1.001 1.001 1.005 1.005 1.005 1.006 1.006 1.086 1.087 1.087 1.087 0.931 0.998 1.009 1.009 1.009 1.009 1.010 0.970 1.010 1.010 1.011 1.011 1.011
0 18.3805 2.8233 9.5400 13.6445 11.8520 4.1875 7.5529 12.3150 4.0006 19.7704 6.3530 4.5221 6.9794 8.1968 13.5898 4.5766 11.1094 7.0672 7.0275 3.2876 10.7986 2.0421 9.0268 2.9669 3.8338 6.8666 0.0252 7.3612 9.1833
– 62 57 219 21 34 50 144 30 134 88 66 1 76 153 42 39 41 47 9 159 24 116 47 17 52 88 50 133 134
**
Total computing time using conventional real-coded genetic algorithm = more than 72 h. Total computing time for the total 362-bus ING load flow solution using enhanced RCGA = 10.38 s. The generators are renumbered as the first thirty buses to facilitate programming in MATLAB. Then, they are revised to their actual number.
Table 5 Comparison of reduction in computation time and storage requirement using the enhanced real-coded genetic algorithm. Type of power system
Matrix density of [Y] (%)
% Reduction in computation time enhanced RCGA (%)
% Reduction in storage requirement enhanced RCGA (%)
14-Bus IEEE 30-Bus IEEE 300-Bus IEEE 362-Bus (ING)
17.34
86
60
7.88
92
75
0.0045
94
98
0.628
98
95
gives us the required accuracy. Here, (w) is chosen to be (0.5) in this research [18,19]. Because of the stochastic nature of the genetic algorithm process, each independent run will probably produce a different number of generations and consequently the computation time and the best amongst these would be chosen. The best of the 10 implementation runs are shown in the tables. For 14-bus IEEE the total computation time was 7.156 s. Table 2 illustrates the power flow solution of the same IEEE test system using RCGA with the method of reduction and restoration as well as sparsity techniques (the proposed method). Since we only retain the generator buses for the GA process, so a single objective function (mismatch active power) is needed because the reactive power for generator buses
Fig. 1. Evaluation process for bus (2), 14-bus IEEE test system using conventional RCGA.
are not specified but calculated as a part of the solution. The total computation time was 0.18 s. Table 3 shows the load flow solution of 30-bus IEEE using the enhanced GA. The computation time for 300-bus IEEE by conventional RCGA was more than 72 h. The CPU time by the proposed method was 17 min without sparsity technique then, with sparsity technique was 50 s. The power flow solution results for the Iraqi National Grid (362 buses) by using enhanced GA are tabulated in Table 4. Since the proposed method implements the complete cycles of the genetic algorithm on the generator busbars only for the first thirty buses of the system, then Table 4 shows the results and number of GA generations for each generator busbar. The voltages of load buses are calculated after restoration of the system, as well as the mismatch active and reactive powers of load buses are calculated. The total computation time with the conventional RCGA method was more than 72 h, whereas the total computation time of the load flow solution of 362-bus ING by using RCGA with the method of reduction and restoration of the system without using sparsity technique was 519 s, while the proposed method (with sparsity technique) was 10.38 s with the same accuracy. Table 5 illustrates the reduction in computational time and storage requirements for different power systems by using the proposed enhanced GA method. The reductions in computation time and storage requirements increase as the matrix density decreases. Fig. 1 shows the evaluation process of the RCGA for bus 2 of the 14-bus IEEE system which is the best of ten implemented runs, the dotted curve represents the minimum cost of the solution (chromosome) which converged at 15 generations and the solid curve represents the average value of the costs amongst generations versus the number of generations. Fig. 2 shows the evaluation process using the enhanced RCGA for the same bus which converged at six generations. 6.2. Load flow solutions of ill-conditioned power systems One of the important criteria of the robustness of the load flow solution methods is their reliability to solve ill-conditioned power systems. The tests were performed on the ‘‘14-bus IEEE’’ typical test system with inserted series capacitors in three lines connecting the buses and shunt capacitors on three buses. The load flow solution is solved by two numerical methods (Gauss–Seidel and Newton–Raphson) [9] as well as the real-coded genetic algorithm to examine its reliability with such systems. The self admittances
310
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
1.5
converged without facing any difficulties. Also, many tests were implemented on different power systems which were used in this research by varying the values of the diagonal elements (imaginary part ‘‘Susceptance’’) of the nodal admittance matrix between ranges small negative, zero, and positive so that to deteriorate the diagonal dominance as ill-conditioned systems in reality by insertion of different values of series and shunt capacitors in branches and busbars respectively, as well as contingent conditions, the enhanced GA solved the load flow problem for these systems without any difficulties. Thus, the proposed enhanced GA is reliable for solving all ill-conditioned power systems [24–27].
cost
1
0.5
7. Conclusions 0
0
5
10
15
20
25
30
35
40
generation Fig. 2. Evaluation process for bus (2), 14-bus IEEE test system using enhanced RCGA.
of the three busbars after inserting series and shunt capacitors in per-unit are:
Bus 1 6:025 þ j11:25;
Bus 4 10:513 þ j7:5;
Bus 5 9:568
j1:21 The load flow solutions by applying these methods for an illconditioned system of the ‘‘14-bus’’ IEEE test system with an accuracy of (0.001 p.u.) are diverging. Table 6 illustrates the results of the power flow solution for the ill-conditioned 14-bus IEEE test system using conventional real-coded genetic algorithm. From the results shown below, we can clearly see that the reactive power mismatches at buses (4) and (5) do not reach the given accuracy even after the full number of the GA generations (1000) is taken. Also, many different values of the diagonal elements of the nodal admittance matrix from large positive values to large negative values including zero value were taken, such that to be not dominant. As well as, many contingent conditions such as single and multiple lines outages, generator or transformer outage, the RCGA failed to solve the load flow problem for such systems. Thus, the conventional real-coded genetic algorithm is unsuitable for solving all ill-conditioned power systems. To explore the performance of the proposed enhanced real-coded genetic algorithm to solve the load flow problem for the ill-conditioned power systems; the enhanced RCGA based load flow solution of the ill-conditioned 14-bus IEEE system mentioned in sub-section (B),
The proposed enhanced RCGA method presented in this paper is very much faster than the conventional genetic algorithm since the system is reduced to the size of that of the generator busbars which for any realistic system is small as we see for the 362-bus Iraqi National Grid, only 30 buses are generator busbars. Also, using sparsity technique makes more reduction in total computation time as well as needs less storage requirements. The objective function (cost function) for the generator buses is only the mismatch active power, so that multi-objective function techniques are not needed. So it can be concluded that the proposed method is suitable for on-line implementation for small and medium-scale power systems and it can be used for a reasonable on-line implementation or planning study for large-scale systems. The proposed method has reliable convergence and high accuracy of solution. The proposed enhanced RCGA can efficiently solve the load flow problem of different types of ill-conditioned power systems while the conventional RCGA as well as many numerical methods have failed to solve such ill-conditioned systems. Whereas the traditional numerical techniques (Gauss–Seidel, Newton–Raphson, Fast decoupled, etc.) use the characteristics of the problem to determine the next sampling point (e.g. gradient, linearity and continuity), the enhanced genetic algorithm makes no such assumptions. Instead, the next sampled point is determined based on stochastic sampling or decision rules rather than on a set of deterministic decision rules. Enhanced genetic algorithms have been used to solve difficult problems with objective functions that do not possess properties such as continuity, differentiability and so forth. Also, whereas the traditional numerical techniques mentioned above use a single point at a time to search the problem space, enhanced genetic algorithm uses a population of candidate solutions for solving the problem, thus, reducing the possibility of ending at a local minimum.
Table 6 Power flow solution (14-bus IEEE) ill-conditioned system with accuracy (0.001 p.u.), ‘‘0.1 MW/MVAr’’ using conventional real-coded genetic algorithm (RCGA). Bus no.
DP (p.u.)
DQ (p.u.)
Voltage magnitude (p.u)
Voltage angle (°)
No. of generations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total computing time:
Slack 0.00042 0.0001 0.00048 0.00089 0.00089 0.00036 0.00022 0.00018 0.00027 0.00095 0.00041 0.00077 0.00029 14.425 s
Slack PV PV 34.561 13.786 PV 0.000060 PV 0.000682 0.000322 0.000223 0.000585 0.000521 0.000962
1.06 1.045 1.01 0.9346 0.965362 1.07 0.99918 1.09 1.056395 1.00163 1.02961 1.021725 1.032889 1.048537
0.00 3.9917 5.3582 7.30252 1.7120 6.30252 14.6541 12.4235 1.44081 9.0031 5.4828 6.67754 11.028 3.4446
– 25 7 1000 1000 21 127 47 172 18 90 48 29 50
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
311
Fig. A-1. Connection lines between regions of (400 KV and 132 KV) grids.
Acknowledgments The authors thank the Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia for its support for making this research. Also, the corresponding author thanks the Department of Electrical Engineering, Engineering College, Baghdad University, Baghdad, Iraq.
Most of the transmission lines are double circuit. This system can be divided into three main regions: i. The northern region. ii. The central region. iii. The southern region.
Iraqi National Grid (ING) has a special quality, which it consists of two main connection grids at the same time:
This is a widely distributed system with very little reserve at the moment, also with many voltage levels which must be taken into consideration in the development of the grid according to geographical distribution of loads and power stations; these levels are:
i. 132 kV connection grids. ii. 400 kV connection grids. ING (under study) contains 599 branches and 362 busbars, most of which are load buses.
i. Connection level, super high voltage grid (400 KV grid), part of (132 KV grid). ii. Transmission level, all the reminder of the (132 KV grid).
Appendix A
312
H. Kubba, H. Mokhlis / Electrical Power and Energy Systems 43 (2012) 304–312
iii. Distribution level, (33 KV grid) and (11 KV grid). There are very few buses of distribution Level (about five buses). Fig. A-1 shows the three regions connection through the two main connection grids (400 KV and 132 KV Grids). The heavy industrial loads lie mainly in the central region with some heavy load scattered around the northern and southern regions. References [1] Lving Malcolm. Pseudo-load flow formulation as a starting process for the Newton–Raphson algorithm. Int J Electr Power Energy Syst 2010;32(8):835–9. [2] Kubba HA, Krishnaparandhama T, Hassan AS. Comparative study of different load flow solution methods. Al-Muhandis, Refereed Sci J Iraqi Eng Soc 1991;107:25–46. [3] Wong KP, Li A, Law TM. Advanced constrained genetic algorithm load flow method. IEEE Proc Generator Transmiss Distrib 1999;146(6). [4] Ippolito L, La Cortiglia A, Petrocelli M. Optimal allocation of facts devices by using multi-objective optimal power flow and genetic algorithms. Int J Emerg Electr Power Syst 2006;7(2):1–19. [5] Bouktir T, Slimani L, Belkacemi M. A genetic algorithm for solving the optimal power flow problem. Leonardo J Sci 2004:44–58. [6] Yin X. Application of genetic algorithm to multiple load flow solution problem in electrical power systems. In: IEEE proceedings of the 32nd conference on decision and control, San Antonio, Texas; December 1993. [7] Yin Xiaodong. Investigations on solving the load flow problem by genetic algorithms. Electr Power Syst Res 1991;22(3):151–63. [8] Kubba HA, Omar R, Soltani J. A multi-objective genetic algorithm for a rapid and efficient load flow solution for electrical power systems. In: Proceedings of the international conference on modelling and simulation, Petra, Jordan; 18– 20 November 2008. p.14–9. [9] Kubba HA. A rapid and more reliable load flow solution method for illconditioned power systems. Eng Technol Refereed Sci J Univ Technol Baghdad, Iraq 1998;17(5):46–59. [10] Vasconcelos JA, Adriano RLS, Vieira DAG, Souza GFD, Azevedo HS. NSGA with elitism applied to solve multi-objective optimization problems. J Microwav Optoelectron 2002;2(6):59–69. [11] Haupt RL, Haupt SE. Practical genetic algorithms. 2nd ed.; 2004.
[12] Kubba HA. An efficient and more reliable second order load flow solution method. Int J Assoc Advance Model Simul Tech Enterprices 2009;46(1):1–19 [Nos. 1–2]. [13] Younes M, Rahli M. On the choice genetic parameter with Taguchi method applied in economic power dispatch. Leonardo J Sci 2006:9–24. issue 9. [14] Krishnaparandhama T, Elangovan S, Kuppurajulu A, Sankaranarayanan V. Fast power flow solution by the method of reduction and restoration. IEE Proc 1980;127:90–3 [Pt. C., No. 2]. [15] Brameller A. Sparisty. Pitman Publishing; 1976. [16] MATLAB: The language of technical computing; 1998.
. [17] Al-Bakri AA. A study of some problems on the Iraqi National Grid and establishing a method algorithm for load flow. M.Sc. thesis. University of Baghdad; 1994. [18] Jain LC, Martin NM. Fusion of neural networks, fuzzy systems and genetic algorithms: industrial applications. LLC: CRC Press; 1998. [19] Ibrahim SBM. The PID controller design using genetic algorithm. A dissertation submit ted to University of Southern Queensland, Faculty of Engineering and Surveying, Electrical and Electronics Engineering; October 2005. [20] Abdul-Haleem GF. A genetic algorithm for manufacturing cell formation. M.Sc. thesis. Mechanical Engineering Department, University of Baghdad; September 2005. [21] Line and Bus Data for 14-Bus IEEE. . [22] Line and Bus Data for 30-Bus IEEE. . [23] Line and Bus Data for 300-Bus IEEE. . [24] Moradi MH, Abedini M. A combination of genetic algorithm and particle swarm optimization for optimal DG location and sizing in distribution systems. Int J Electr Power Energy Syst 2012;34(1):66–74. [25] Eidiani Mostafa. A reliable and efficient method for assessing voltage stability in transmission and distribution networks. Int J Electr Power Energy Syst 2011;33(3):453–6. [26] Mohammadi M, Hosseinian SH, Gharehpetian GB. GA-based optimal sizing of microgrid and DG units under pool and hybrid electricity markets. Int J Electr Power Energy Syst 2012;35(1):83–92. [27] Mokhlis Hazlie, Li Haiyu. Non-linear representation of voltage sag profiles for fault location in distribution networks. Int J Electr Power Energy Syst 2011;33(1):124–30.