The transmission expansion planning methods were initially focused only on the least cost parameter. In last few years, due to the availability of the fast and accurate software tools, the transmission expansion planning has experienced a quality improvement. Many optimization algorithms with the greater level of uncertainty have been introduced.
This paper is mainly focused on the meta-heuristic optimization of transmission expansion planning. It discusses the design and implementation issues of Leap Frog optimization method in transmission expansion planning while modifying the method according to common electrical transmission constraints. The presented paper also discusses the working methodology and empirical results of Leap Frog method for optimizing of the objective function with quick convergence.
Index Terms- TEPP, Meta-Heuristic Approach, Frog Leap Algorithm.
I. INTRODUCTION
Inspite of deregulation in the electricity market throughout the world, the transmission network remained a large non-competitive sector because it requires heavy investment in huge infrastructure. The generated electric power needs to be delivered to the consumers through a network owned by different entities. If the transmission line limits or the maximum power flow through the line is surpassed, then the cheap generation (at far end) might become costly, while the new generation which is nearer and less costly would be a better alternative to limit the line flow.
Transmission expansion planning is a constrained optimization technique which gives an optimal strategy of 'when, where and how many lines' are to be added to satisfy future load-growth as well as reliability criterion. Planning of transmission network expansion begins with the development of generation and load forecasts scenarios, which provides an eventual necessity to strengthen and broaden the existing network. To satisfy the electricity service condition, a lucid plan should be developed that has coherence with generation and demand in the network. All this planning should aim at satisfying technical and operating conditions.
The mathematical model of TEPP is quite complex as it has non-linear nature with large number of variables and parameters involved. Huge investments are involved in modernizing and expanding the network with the integration of newly added generation. With the fragmentation of the electricity industry, the Transmission expansion planning (TEP) is also decoupled with Generation Expansion Planning (GEP). This has changed the fundamental nature of the TEP problem [1] that is, now much more than in the past, influenced by uncertainties affecting. In the recent scenario the transmission system have to provide open access to different entities for power trading and the deficit transmission capacity may limit power trading.
Several methods have been adopted to solve the TEPP both in regulated as well as deregulated environment. There are many mathematical models that are traditionally available to solve the transmission expansion problem from the cost minimization standpoint. These models may be based on linear and mixed-integer programming [2-5], dynamic programming [7], heuristic methods [8], genetic algorithms [9], and game theory [11-13] etc. With the introduction of pool-based markets and bilateral contracting, the new transmission expansion models propose social welfare maximization instead [14-15].
The optimization of transmission is very important to the efficiency and economics of overall electric grid. Electrical network expansion is often difficult because of the complex solution space. Several researches deployed optimization systems based on the efficient behavior of species such as ants, bees, birds and frogs, while seeking faster and more robust solutions. The first evolutionary-based technique introduced in the literature was the genetic algorithm. In an attempt to reduce processing time and improve the quality of solutions while avoiding local optima, various improvements in genetic algorithm and other evolutionary methods have been proposed in last decade.
The shuffled frog-leaping (SFL) algorithm [16-17], is one of the most promising technique of optimization, which could deal with common optimization problems like slow convergence and local optimum.
This paper is organized as follows: the second part gives an overview of the network expansion problem; third part gives detailed description of Frog Leap algorithm, fourth part shows the implementation and finally the results and conclusion critically examines the testimony.
II. NETWORK EXPANSION MODEL
The Transmission expansion planning problem deals with the decision of finding optimum routes between generation and load, satisfying the load demand under both normal and contingency condition, while satisfying the conventional least cost criterion. Further, the transmission expansion cost allocation becomes a critical and pragmatic issue of deregulated electricity industry. The network expansion planning problem can be solved by various optimization techniques, each having its own pros and cons.
For adding a new candidate to the existing system, a number of techniques may be used as in literature. The expansion problem is formulated as a cost minimizing problem subjected to technical and economical constraints while considering the N-1 contingency.
The aim is to minimize the total cost C which is the investment cost for the new candidates. The network cost is calculated as the cost of candidates, where candidate cost is expressed in cost per unit length. The network cost for a selected topology is calculated as:
Ctotal = (1)
where,
Dk is the length of transmission line of the candidate,
k is the set of candidates,
Lk is the transmission type of the candidate,
Ck(Lk) is the investment cost per km for kth pipe with type Lk,
η is the number of lines to be added in corridor ij, i.e. total lines between corridor ij minus existing lines in the same corridor.
The above formulated problem can be solved by various optimization techniques, each having its own pros and cons. We have assumed that the line cost for all types to be one monitory unit per km length. Several constraints are imposed in mathematical optimization to ensure that the solutions are in-line with the requirements. The constraints imposed are as follows:
(2)
(3)
where and are the voltage phase angles of buses i and bus j;
Bij is the imaginary part of the element ij of the admittance matrix,
PGi is the power generation at bus i ,
PDi is the power demand at bus i, and n is the set of system buses.
The index m in third equation shows the contingency parameters and variables.
c is the set of contingencies.
n is the number of buses.
For the transmission lines, the power transfer should not violate its rating (limits) during both normal and contingency conditions, therefore
(4)
K ϵ Lc + Le, (5)
where, Le is the set of existing lines,
Lc is the set of candidates,
Pk is the line rating during normal condition,
Again the index m forms second equation which is under contingency condition.
III. SHUFFLED FROG LEAP ALGORITM
Shuffled Frog Leap Method is basically a metaheuristic approach for solving optimization problems. It was introduced by Eusuff et. al for optimizing water distribution system, which is basically a population-based cooperative search (inspired by natural memetics) [16]. They have used the memetic evolution in the form of influence of ideas among individuals through a local search as in swarm optimization. This shuffling strategy allows the exchange of information in local entities that tends towards a global optimum [16].
In Shuffled Frog Leap method, the population consists of a set possible solutions (known as frogs), which are divided into subsets (known as memeplexes). A local search is performed within all memeplexes, each representing different frog cultures. The individual frog (belongs to a memeplex) holds own ideas, which can be influenced by the ideas of frogs in their memeplexes. These ideas evolve through a process of memetic evolution while performing local search. When the steps of memetic evolution reached to the predefined limit, these ideas are exchanged (if required) among memeplexes by a shuffling process [18]. This process of local search and shuffling iterates till the convergence intimation is signaled [17].
The process could be explained as follows: at the first step, an initial population of possible solutions (frogs) is generated randomly. Then these solutions (frogs) are to be sorted in a descending order of their fitness value. The entire population is then divided into r memeplexes wherein each memeplex containing s frogs. The process of dividing the frogs in memeplexes is such that, the frog at first position goes to the memeplex one, the next frog goes to the memeplex two, and like-wise frog r goes to the rth memeplex, and frog r+1 goes back to the first memeplex and so on. The best solution and worst solution are to be identified and terms as Fb and Fw, (best frog and worst frog) respectively in each memeplex and the global best frog (best fitness globally) is to identified and termed as Fg. Then, the Frog Leap algorithm applies the procedure to improve the fitness of the worst frog in each cycle.
The improvement in the position of the worst frog is to be calculated as follows [17]:
Change in frog position:
Xi = rand Ã- (Fb - Fw ) (6)
New position: Xi+1 = current frog position Xi +Di (7)
where,
-Dmax ≤ Di ≤ Dmax, (8)
where rand is a random number in between 0 and 1, maximum allowed change in a frogs position is Dmax. If a better solution is yielded, it replaces the worst frog. If not, then the calculations in above equations are repeated by replacing memeplex's best by the global best frog (i.e. Fb is replaced by Fg). If there is no improvement, then again a new solution is to be generated randomly for replacement. The process repeats for pre-specified number of iteration [16].
The main drawback of SFL algorithm is slow convergence, which is closely related to the lack of adaptive acceleration term in the position updating formula. In equation (6), the term rand determines the movement in step-sizes of frogs through the Fb and Fw positions. In the standard SFL, these step-sizes are random numbers between 0 and 1 for all frogs. In each cycle, the value of the objective function is a criterion that presents the relative improvement of a frog movement with respect to the previous one. Thus the difference between the calculated values of the objective function in consequent iterations can represent the frog acceleration.
IV. SOLUTION ALGORITHM
Shuffled Frog Leap algorithm is an approach which combines the advantages of genetic-based Memetic Algorithms and the social behavior-based Particle Swarm Optimization. The proposed methodology includes the use of SFLA for TEPP, following are the steps involved in solving TEPP using SFLA:
Step 1: Specify various parameters for DC load flow, generation and load at each bus, line parameters, number of candidate lines, and number of parallel paths.
Step 2: Initialize the population of frogs randomly, X = [X1, X2, ----------------------, XN]
Here for n dimensional problem, each frog is represented by n decision variables like Xi = (Xi1, Xi2, ------------ Xin). In TEPP, the ultimate output is the number of corridors in each path and hence the frog population consists of the candidates with different parameters.
The matrix is initialized randomly and each frog is generated after satisfying the power flow constraints, so that the optimization process (local search as well as global search) starts within the feasible region.
Step 3: Initialize other parameters of SFLA like number of frogs n, number of memeplexes m, so the number of frogs per memeplex are n/m.
Step 4: At this stage, the fitness of each frog is calculated. This fitness value actually represents the total cost for the particular sets of candidates.
Step 5: The frogs are arranged as per their fitness value in the descending order, and then are divided into m memeplexes, such that frog with best fitness goes to 1st memeplex, second ranked frog goes to 2nd memeplex and so on till mth ranked frog goes to mth memeplex and then again the cycle is repeated.
Step 6: Within each memeplex the frog with the best fitness Fb, and the worst fitness frog Fw is identified, further, the frog with best fitness in whole population of frogs is identified as global best Fg.
Figure 1: Flowchart of SFLA [23]
Step 7: At this stage, the individual memeplex evolution takes place, i.e. local search (generally named as memetic evolution), wherein each group of frogs evolves as an independent culture.
During this evolution process the position of worst frog is improved by using the equation number (6) and (7).
Set the memeplex iteration count to i =1
Find frogs with Fb and Fw.
Change the position of worst frog by using frog leaping rule described by equation (7).
Calculate the fitness for this frog with new position.
If this improves the fitness then replace the old frog with the new one, and go to Sub-Step 7.8.
Otherwise the frog leaping rule is applied by replacing the memeplex best with the global best in equation 1.
Calculate the fitness, if improves then replace with the new one.
Still if the frog fitness cannot be improved, then a new frog is randomly generated to replace the worst frog. This is done to prevent the defective meme from creating new frogs with decreasing fitness value, since this will prevent the movement towards the optimal solution.
Check i ≤ Im (memeplex iteration count), Go back to Sub-Step 7.1.
7.10 Repeat the process for all memeplexes.
Step 8: After the complete local search (memeplex evolution), the complete frog population of all the memeplexes is shuffled. This step enables the global exchange of information. This further moves the search region of the memeplexes towards the global best position. Then go to Step 5.
Step 9: Check for termination.
9.1 If number of shuffling iterations j ≤ specified iterations, or
9.2 Convergence criterion is reached.
[|Xw new | - | Xw old |] < Є, (9)
where Є is the convergence tolerance
Step 10: Stop.
V. RESULTS AND DISCUSSION
A. Test System:
The above described algorithm is applied to Graver 6-bus system as mentioned by Graver [19]. Figure 2 shows modified system with demand forecasts, increase in load, generation and the addition of new bus.
The data for the test system is mentioned in Table 1(Bus Data) and Table 2 (Network Data).
TABLE 1GENERATION AND LOAD DATA
BUS
LOAD
GENERATION
PD (p.u.)
QD (p.u.)
PG (p.u.)
1
0.240
0.116
1.130
2
0.720
0.348
0.500
3
0.120
0.058
0.650
4
0.480
0.232
-
5
0.720
0.348
-
TABLE 2 NETWORK DATA
Line No.
BUS
R (p.u.)
X (p.u.)
Capacity limits(p.u.)
Path length (km)
From
To
1
1
2
0.1
0.40
1.0
400
2
1
4
0.15
.60
0.8
600
3
1
5
0.05
0.20
1.0
200
4
2
3
0.05
0.20
1.0
200
5
2
4
0.10
0.40
1.0
400
6
3
5
0.05
0.20
1.0
200
7
1
3
0.095
0.38
1.0
380
8
2
5
0.0775
0.31
1.0
310
9
3
4
0.1475
0.59
0.8
590
10
4
5
0.1575
0.63
0.8
630
The Network data mentioned above shows the existing line data as well as candidate line data.
Table 3 and 4 below shows the construction cost for 230 KV and 400 KV line for single and double line.
Untitled.png
Figure 2 Network with future load and generating unit [7].
TABLE 3: CONSTRUCTION COST OF 230KV
Number of line circuits
Fixed cost of line construction
(Ã-103 dollars)
Variable cost of line construction
(Ã-103 dollars)
1
546.5
45.9
2
546.5
63.4
TABLE 4: CONSTRUCTION COST OF 400 KV
Number of line circuits
Fixed cost of line construction
(Ã-103 dollars)
Variable cost of line construction
(Ã-103 dollars)
1
1748.6
92.9
2
1748.6
120.2
B. Setting the SFLA parameters:
TABLE 5: PARAMETERS INITIALIZATION
SFLA Parameters Initialization
Number of frogs
Number of memeplexes
Frogs per memeplex
Memeplex evolution count
Total generation count
C. Results:
The results obtained for adding the candidates for Graver 6-bus system considering single voltage level is shown in Table 6.
The expansion cost as mentioned above in table 3 and table 4 is considered for calculations but for comparison of results with other approaches, cost is assumed to be one monetary unit per km (whatever the type of system). Here 1 for voltage level represents only one voltage either 230 KV or 400 KV. This simplification takes into account the difference between the costs adopted for solution in different scenarios.
TABLE 6: EXPANSION MODEL, SCENARIO 1
Corridors
No. of circuits
Voltage level
2-6
2
1
3-5
1
1
4-6
2
1
2-3
1
1
Untitled.png
Figure 3. Scenario 1: Expansion plan with SFLA
The expansion plan in scenario 1 (Figure 3), considering generation rescheduling i.e., each generator output can vary from 0 to maximum generation. The dotted lines represent newly added candidates. Now, the test results obtained using SFLA shows the total cost of 130 units, which is same as obtained by Contreras [20].
TABLE 7: EXPANSION MODEL, SCENARIO 2
Corridors
No. of circuits
Voltage level
2-6
4
1
3-5
1
1
4-6
2
1
Untitled.png
Figure 4. Scenario 2: Expansion plan with SFLA
Scenario 2, represents the case as mentioned by Graver [19]. The condition considered is without rescheduling. The expansion cost obtained is 200 monetary units, same as obtained by Graver.
Various other scenarios are used to test the proposed algorithm, like under N-1 contingency criterion, i.e. the condition when either one line is removed or the generation at any bus is reduced by 40%. Only some scenarios are discussed out of the various experiments conducted, hence only one scenario is discussed with contingency and rescheduling.
TABLE 8: EXPANSION MODEL, SCENARIO 3
Corridors
No. of circuits
Voltage level
2-6
1
1
3-5
2
1
4-6
3
1
2-3
2
1
The total expansion cost for scenario 3 is 200 units with rescheduling, equal to expansion plan without rescheduling and without contingency. This reflects the reduction in total cost.
D. Comparison:
Comparison of SFLA with other algorithms is shown in table 9. Apart from SFLA other results are obtained from [21].
TABLE 9: COMPARISION OF AI TECHNIQUES AS APPLIED TO TEPP
Method
Total Cost
SFLA (w/o contingency)
130
SFLA (with contingency)
200
TS
200
GA
200
ANN
200
GA/TS
200
GA/GA
200
ANN/GA
200
ANN/TS/GA
200
Linear Model
200
A comparison of various AI techniques is presented in [17]. The authors have compared GA, MA, PSO, ACO and SFLA. The processing time of SFLA and the rate of approaching to the desired optimal solution is the best of among other AI techniques, i.e. the success rate, of SFLA is much better than GA and similar to PSO [17].
VI. CONCLUSION
In this paper a modern AI approach SFLA is discussed for strategically configuring the optimization mechanism for transmission expansion planning. The SFLA as applied to TEPP produces faster and better results. Mathematical Equations are formulated and modified accordingly. Three different scenarios are discussed in detail. The results have shown the out-performance of said algorithm on the standard data of Graver six-bus system.
VII. REFERENCES
[1] C. W. Lee, S. K. K. Ng, J. Zhong, F. F. Wu, "Transmission Expansion Planning From Past to Future", in Proceedings of the IEEE 2006 Power Systems Conference and Exposition, Atlanta, pp. 257 - 265.
[2] L. L. Garver, "Transmission network estimation using linear pro-gramming," IEEE Trans. Power App. Syst , vol. PAS-89, no. 7, pp.1688-1697, Sep./Oct. 1970.
[3] R. Villasana, L. L. Garver, and S. J. Salon,"Transmission network plan-ning using linear programming, " IEEE Trans. Power App. Syst., vol. PAS-104, no. 1, pp. 349 -356, Feb. 1985.
[4] R. Romero and A. Monticelli, "A hierarchical decomposition approach for transmission network expansion planning," IEEE Trans. Power Syst. , vol. 9, no. 1, pp. 373 -380, Feb. 1994.
[5] N. Alguacil, A. L. Motto, and A. J. Conejo, "Transmission expansion planning: A mixed-integer LP approach, " IEEE Trans. Power Syst. , vol. 18, no. 3, pp. 1070 -1076, Aug. 2003.
[6] S. Binato, M. V. F. Pereira, and S. Granville, "A new Benders decom-position approach to solve power transmission network design prob-lems," IEEE Trans. Power Syst. , vol. 16, no. 2, pp. 235 -240, May 2001.
[7] Y. P. Dusonchet and A. H. El-Abiad, "Transmission planning using discrete dynamic optimization, " IEEE Trans. Power App. Syst. , vol. PAS-92, no. 2, pp. 1358 -1371, Apr. 1973.
[8] E. J. de Oliveira, I. C. da Silva, J. L. R. Pereira, and S. Carneiro, Jr., "Transmission system expansion planning using a sigmoid function to handle integer investment variables," IEEE Trans. Power Syst. , vol. 20, no. 3, pp. 1616-1621, Aug. 2005.
[9] R. A. Gallego, A. Monticelli, and R. Romero, "Transmission system expansion planning by extended genetic algorithm, " Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 145, no. 2, pp. 329 -335, May 1998.
[10] R. Romero, R. A. Gallego, and A. Monticelli, "Transmission system expansion planning by simulated annealing," IEEE Trans. Power Syst. , vol. 11, no. 1, pp. 364 -369, Feb. 1996.
[11] J. Contreras and F. F. Wu, "Coalition formation in transmission expan-sion planning, " IEEE Trans. Power Syst. , vol. 14, no. 3, pp. 1144 -1152, Aug. 1999.
[12] J. Contreras and F. F. Wu, "A kernel-oriented coalition formation algo-rithm for transmission expansion planning," IEEE Trans. Power Syst. , vol. 15, no. 4, pp. 919 -925, Nov. 2000.
[13] J. M. Zolezzi and H. Rudnick, "Transmission cost allocation by coop-erative games and coalition formation," IEEE Trans. Power Syst. , vol. 17, no. 4, pp. 1008 -1015, Nov. 2002.
[14] G. B. Shrestha and P. A. J. Fonseka, "Congestion-driven transmission expansion in competitive power markets, " IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1658-1665, Aug. 2004.
[15] J. Contreras, V. Bósquez, and G. Gross, "A framework for the analysis of transmission planning and investment," in Proc. 15th Power Systems Computation Conf. ,Liége, Belgium, Aug. 22-26, 2005.
[16] MM. Eusuf, KE. Lansey, "Optimization of water distribution network design using the shuffled frog leaping algorithm". J Water Resour Plan Manage, Vol 129(3), 2003, pp. 210-225.
[17] Emad Elbeltagi,Tarek Hegazy, Donald Grierson, "Comparison among five evolutionary-based optimization algorithms", Advanced Engineering Informatics 19 (2005) 43-53
[18] S. Y. Liong, Md. Atiquzzaman., Optimal design of water distribution network using shuffled complex evolution. J Inst Eng, Singapore, Vol 44(1), 2004, pp. 93-107.
[20] J. Contreras and F. F. Wu, "Coalition formation in transmission expan-sion planning, " IEEE Trans. Power Syst. , vol. 14, no. 3, pp. 1144 -1152, Aug. 1999.
[21] Tawfiq Al-Saba, Ibrahim El-Amin, "The application of artificial intelligent tools to the transmission expansion problem", Electric Power Systems Research 62 (2002) 117/126
[22] Liong S-Y, Atiquzzaman Md. Optimal design of water distribution
network using shuffled complex evolution. J Inst Eng, Singapore 2004;44(1):93-107.
[23] Optimal Placement and Sizing of DG in Radial Distribution Networks Using SFLA , E. Afzalan, M. A. Taghikhani, M. Sedighizadeh. International Journal of Energy Engineering 2012, 2(3): 73-77
[24] S.Jaganathan, Dr.S.Palaniswami, C.Sasi Kumar R.Muthu Kumaran, "Multi Objective Optimization for Transmission Network Expansion Planning using Modified Bacterial Foraging Technique", International Journal of Computer Applications (0975 - 8887) Volume 9- No.3, November 2010
[25] Tawfiq Al-Saba, Ibrahim El-Amin, "The application of artificial intelligent tools to the transmission expansion problem", Electric Power Systems Research 62 (2002) 117/126
[26] Hossein Seifi. "Network Expansion Planning, a Basic Approach", Power Systems, 2011
[27] SebastiÁn de la Torre. "Transmission Expansion Planning in Electricity Markets", IEEE Transactions on Power Systems, 02/2008
[28] Emad Elbeltagi. "A modified shuffled frog-leaping optimization algorithm: applications to project management", Structure & Infrastructure Engineering Maintenance Management Life-Cycle Design & Performance, 3/1/2007
[29] Xia Sun. "An SFL-Based Multicast Routing Optimization Algorithm", 2009 International Workshop on Intelligent Systems and Applications, 05/2009
[30] Jalilzadeh, . "A GA Based Transmission Network Expansion Planning Considering Voltage Level, Network Losses and Number of Bundle Lines", American Journal of Applied Sciences, 2009.
[31] R. Venkata Rao. "Overview", Springer Series in Advanced Manufacturing, 2011
[32] Ming-Huwi Horng. "Multilevel image thresholding by using the shuffled frog-leaping optimization algorithm", The 16th North-East Asia Symposium on Nano Information Technology and Reliability, 10/2011
[33] Gomez-Gonzalez, M., and F. Jurado. "Personalized e-learning using shuffled frog-leaping algorithm", Proceedings of the 2012 IEEE Global Engineering Education Conference (EDUCON), 2012.
[34] Wang, N.. "Fast three-dimensional Otsu thresholding with shuffled frog-leaping algorithm", Pattern Recognition Letters, 20101001
[35] Venkatesan, T., and M.Y. Sanavullah. "SFLA approach to solve PBUC problem with emission limitation", International Journal of Electrical Power & Energy Systems, 2013.
[36] Soliman Abdel-Hady Soliman. "Optimal Power Flow", Energy Systems, 2012
[37] Zhang, Hui, Gerald Thomas Heydt, Vijay Vittal, and Hans D. Mittelmann. "Transmission expansion planning using an ac model: Formulations and possible relaxations", 2012 IEEE Power and Energy Society General Meeting, 2012.
[38] Mohammad-Taghi Vakil-Baghmisheh. "A modified very fast Simulated Annealing algorithm", 2008 International Symposium on Telecommunications, 08/2008
[39] Junwan Liu. "Multiobjective optizition shuffled frog-leaping biclustering", 2011 IEEE International Conference on Bioinformatics and Biomedicine Workshops (BIBMW), 11/2011
[40] Jahani, R.. "Optimal DG Allocation in Distribution Network Using A New Heuristic Method", Australian Journal of Basic & Applied Sciences/19918178, 20110501
[41] Madinehi, Nazli, Kiarash Shaloudegi, Mehrdad Abedi, and Hossein Askarian Abyaneh. "Optimum design of PID controller in AVR system using intelligent methods", 2011 IEEE Trondheim PowerTech, 2011.
[42] Suresh Chittineni. "A Modified and Efficient Shuffled Frog Leaping Algorithm (MSFLA) for Unsupervised Data Clustering", Communications in Computer and Information Science, 2011
[43] Mahdavi, M.. "DCGA based evaluating role of bundle lines in TNEP considering expansion of substations from voltage level point of view", Energy Conversion and Management, 200908
[44] Jalilzadeh, Saeid. "PSS and SVC Controller Design Using Chaos, PSO and SFL Algorithms to Enhancing the Power System Stability", Energy & Power Engineering/1949243X, 20110501
[45] Thai-Hoang Huynh. "A modified shuffled frog leaping algorithm for optimal tuning of multivariable PID controllers", 2008 IEEE International Conference on Industrial Technology, 04/2008
[46] Babak Amiri. "Application of shuffled frog-leaping algorithm
