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2010 IREP Symposium- Bulk Power System Dynamics and Control – VIII (IREP), August 1-6, 2010, Buzios, RJ, Brazil

Parallel Genetic Algorithm to Tune Multiple Power System Stabilizers in Large Interconnected Power Systems

Antonio L. B. do Bomfim Glauco N. Taranto, Djalma M. Falcao,

Sergio L. Escalante ELETROBRAS COPPE/UFRJ

Abstract This paper presents a Power System Stabilizer (PSS) tuning procedure based on a Parallel Genetic Algorithm (PGA). The procedure is applied to the simultaneous tuning of 61 PSSs, using their actual models in the entire Brazilian Interconnected Power System (BIPS) modeled with 3849 buses. Heavy, median and light loading conditions were considered during the tuning process, to ensure control robustness. A PGA with small population sizes distributed in various CPUs is utilized to accelerate the searching for the solution. The PGA utilizes the genetic operator called migration. Real number chromosome representation, arithmetic crossover, variable exponential mutation and variable search space are some techniques exploited in the paper.

Index Terms — Small-signal stability; Power System Stabilizer; Parallel Genetic Algorithm; Modal Analysis; Arithmetic crossover, Pareto front. Introduction As electric power system evolves, it becomes more complex. Nowadays, this complexity is related to new kinds of generation like biomass, wind, and solar generators, long-distance DC and/or AC transmission systems, etc, subject to hydraulic and weather conditions. Furthermore, the necessity to operate the systems near their physical limits is being reported in many electrical power systems. To circumvent this reality one may be forced to rely on well tuned control systems, like the Power System Stabilizers (PSS) for instance. The addition of new damping sources [1] - [2] together with more stressed operation in present interconnected power grids trigger the importance for methods that can handle an overall coordination for the system controllers. Conventional design approaches, like decouple and sequential loop closure utilized in [3], cannot properly handle a truly coordinated design. In contrast the papers [4] and [5] presented a method for coordinated tuning of 22 power system damping controllers using Genetic Algorithm (GA) [6] and [7]. In some sense, this paper is an extension of paper [4], where a Parallel Genetic

Algorithm (PGA) using 64 CPUs is applied for the tuning of 61 PSSs in the Brazilian system. GA is a technique for searching the best solution of a problem using intensive combinatorial tentative solutions. It combines and selects chromosomes by genetic operators namely crossover, mutation, elitism and selection, to create a new generation of tentative solutions (newer chromosomes). It is based on Darwin’s Theory where the best individual survives and spreads its genes to the offspring. The PGA implemented in this work is customized to consider variable searching space, migration and variable mutation rate. The variable searching space is performed after a specified number of generations, becoming larger or smaller according to some defined criteria. It improves PGA search to reach the best solution. Initial mutation rate is reset at each change of the searching space. Migration operator is performed after a new space is created. In the present work we validate the proposed formulation into a realistic power system problem. The method is based on the optimization of a function related to the global minimum damping ratio and the minimum interarea damping ratio, related to the closed-loop spectrum, constrained in the controller parameter space. The robustness of the controllers is taking into account during the tuning process, simply by considering a prespecified set of system operating conditions into the objective function. A decentralized coordinate design is performed as the controller channels are closed simultaneously with all the cross-coupling signals amongst controllers not taken into consideration. The PGA was applied to the Brazilian Interconnected Power System (BIPS), with approximately 3800 buses, to tune 61 PSSs using their actual modeling structure. Heavy, median and light loading conditions were considered during the tuning process. Attention was paid to the correct number of generating units in service, as the loading conditions change. Subtransient rotor effects are modeled for 92 generators. The open-loop linearized dynamic representation of the system ended up with a system matrix with 1516 state variables.

Problem Formulation Power system damping controllers are usually designed to operate in a decentralized way. Input signals from remote sites are not considered timely and reliably enough with today’s technology. The advent of Phasor Measurement Units (PMU) may change this scenario in the near future. The robustness of the controllers is ensured by considering the performance of the control system for several different operating conditions. The choice of these conditions is based on experience and simulation studies. Tuning of power system damping controllers uses a small-signal model represented by the well-known state space equations (1).

( ) ( ) ( )( ) ( ) ( )

.x t Ax t Bu ty t Cx t Du t

= += +

(1)

Where x is the vector of the state variables, such as the machine speeds, machine angles, and flux linkages; u is the vector of the input variables, such as the control signals, y is the vector of the measured variables, such as bus voltages and machine speeds; A is the power system state matrix; B is the input matrix; C is the output matrix; and D is the feedforward matrix. Stability of (1) is determined by the eigenvalues of matrix A. Figure 1 shows the damping control loop via the excitation system. AEG(s) represents the combined transfer function for the automatic voltage regulator (AVR), the exciter and the generator, and PSSSC(s) and PSSGA(s) when combined represent the PSS(s) as given by (2).

Figure 1 – Close-loop setup. ( ) ( ) ( )GA SCPSS s PSS s PSS s= ⋅ (2)

One should note that the decomposition of the PSS structure in two transfer functions, is made to preserve the actual structure of the PSSs modeled in the official database [8] utilized in the studies of the BIPS. PSSSC(s) involves all signal conditioning filters, such as ramp-tracking filter, washout, reset functions, etc, and it can receive multiple input signals, as it is the case in many existing PSSs in the BIPS. Figure 2 is an expanded view of Figure 1.

+ EXC(s)AVR(s) GEN(s)

LEDLAG(s)K SignalConditioning

Vpss

Vudc

PSSSC(s)PSSGA(s)

Vref

AEG(s)

y

Vt

+-

+

Figure 2 - Expanded view of the closed-loop setup.

On the other hand, PSSGA(s) has a fixed structure shown in Figure 3, and represents the part of the PSS where the PGA will tune.

K

PSSGA(s)

12

11

lT sT s

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

++

y

Figure 3 - Block diagram of PSS to be tuned by the PGA. The expression that defines the PSSGA (s) to be tuned is shown in (3).

1

1 2

11

( ) , , ,...,

li

iGA li

i i

s

PSS s K i pis

αω

ω α

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠= =⎛ ⎞

+⎜ ⎟⎜ ⎟⎝ ⎠

(3)

Where l is the number of lead-lag blocks in the model, and p is the number of PSSs to be tuned. The parameters of each controller to be determined are iK , iα and iω . Figure 4 shows one example of a PSS model represented in the BIPS. This PSS has two-input signals that synthesize the integral of the acceleration power, has a fourth-order ramp-tracking filter and a washout filter.

Figure 4 - Example of how to consider a real PSS model

to create the open-loop state-space matrices. To form the open-loop state-space matrices we consider the existing gain equal to zero and the lead-lag blocks, for phase compensation, disconnected from the controller. This resource is readily available in PacDyn program [9], where the transfer function considered to creating the state matrices is given by ref udcV V .

The decentralized control design requires a control law such that the closed-loop system is stable and has no eigenvalues with less damping than minimum specified damping ratio minς in all nc operating conditions. Formulation as an Optimization Problem Experience showed us that the initial objective function of maximize the minimum eigenvalue damp [4], was always constrained by local electromechanical modes, impairing improvements to the damping of interarea modes. To mitigate this problem, the objective function was stratified according to the frequency of the electromechanical modes. A double weigh was given to the interarea modes, and a weighted sum of minimum damping factors of interarea and local modes became a new objective function (4). ( )minMax Min 0.33 0.67 interarea nc

Max F damp damp⎡ ⎤= +⎢ ⎥⎣ ⎦ (4)

Where nc is the number of operating condition; mindamp is the minimum damping ratio of the local electromechanical modes, and interareadamp is the minimum damping ratio of the interarea modes. The considered threshold frequency bounds separating the interarea set of modes has a lower bound of 0.25 rad/s and an upper bound of 5 rad/s. The lower bound was considered to exclude modes related to the governors. Subject to (5),

min max

min max

min max

i i i

i i i

i i i

K K Kα α αω ω ω

≤ ≤≤ ≤≤ ≤

(5)

This modification yielded much better solutions for the BIPS, in which three important interarea modes are present, namely South/Southeastern mode, North/ Northeastern mode and North/Southern mode. The PGA was applied to the BIPS (about 3800 buses) to tune 61 PSSs using actual models. It was supposed that the others PSS are well tuned. Heavy, median and light loading conditions were considered during the tuning process. Attention was paid to the correct number of generating units considered according to the loading condition. In terms of the number of lead-lag blocks, there are 9 PSSs with one block and 52 PSSs with two blocks to be tuned. There are only three kinds of feedback stabilizing signals used as inputs in the set of 61 PSSs to be tuned. They are electric power signal used in 10 machines, accelerating power signal used in 2 machines

and integral of accelerating power signal used in 49 machines. Parallelization of GA The solution to the optimization problem defined in (4) and (5) can be obtained using a PGA [10] . PGA Elements Fitness Function: The fitness function ff used by PGA is defined in (4), where ff=max F. Parameter encoding and limits: The controller parameters are encoded as real number string using concatenated multiparameter representation. The PGA is customized to consider variable searching space. The upper and lower limits change along the searching process, and they depend on the size of the searching region. After 10 generations the limits change to reduce the region, and after 5 generations the limits change to expand the region [11]. The variable searching space is performed every 5 or 10 generations, becoming smaller or larger according to some defined criteria. The enlargement of the searching space is performed every 10 generations and it helps to introduce new values to the searching-space bounds. The reduction of the searching space is performed every 5 generations and it helps to fine tune the best solutions found during the enlarged-space search. Genetic Operators: Selection was performed using the tournament scheme. An elitism strategy was used, i.e., the best individual of current population is guaranteed to be present in the next population by replacement of the worst individual of next generation by the best individual of the previous one. Arithmetic uniform crossover operator as described in [11] and [12] was used with 0.85 for crossover rate, 0.5 for mask rate. The factor to change the value position is randomly chosen in the interval [0.0, 1.0]. Mutation was implemented using an exponential decreasing mutation rate, which starts at a specified maximum value and decreases exponentially until it reaches a given minimum value, and return to initial value after each change in the searching space. In this genetic operator the randomly selected gene is replaced by another real number randomly selected within its bounds. The exponential mutation rate used was 0.02 at the beginning and truncated at 0.004. The PGA utilizes the genetic operator called migration [10]. The migration operator selects the best chromosome of each CPU and spread them to the others CPUs in the parallel computational framework, as shown in Figure 5.

Figure 5 - Simple example of parallelization. Depending on the computer power in hand, the number of CPUs may be greater than the number of individuals in the population (Not unusual nowadays!). In such situations the population in each CPU would be the same. To avoid this problem, a smaller subset from the best chromosomes needs to be chosen according to some ranking criterion. In this work, we have chosen only the 4 best individuals among the best 64. This strategy was named elitism of migration. Migration operator is performed after the search space is changed to avoid creating the same region in each CPU. PGA allows small populations in each CPU, but the total individuals that PGA will combine, is the sum of individuals in all CPUs, as shown in (6).

1

NCPUNIPGA NICPU= ∑ (6)

Where NIPGA is the number of all individuals in the PGA; NCPU is the number of CPUs used; and NICPU is the number of individuals in each CPU. In this work it was used NCPU=64 CPUs and NICPU=20 individuals, yielding NIPGA=1280 individuals. The stopping rule is the maximum number of generations. The number of QR eigenvalue calculation method [13] performed until the end of the search is given by (7). *NQR NIPGA NGER= (7) The maximum number of generations was 160, then NQR = 204,800 QRs performed by the PGA to reach the best solution. Each processor performs 3,200 QR’s. The time to perform each QR is 83 seconds. The computer used to perform the parallel computation is the ALTIX ICE having 64 CPUs Quad Core Intel Xeon 2.66Ghz 256 cores from Center for Service in High Performance Computing (NACAD) [14]. Simulation results To verify the performance of the proposed methodology the BIPS was used. It is comprised of 3849 buses, 5519 transmission lines and transformers and 202 generators. Heavy, median and light loading operating conditions

from September 2008 were considered. An HVDC link and generators with capacity lower than 40 MVA were not dynamically modeled. Active power loads are modeled with a mixed percentage of constant current and constant impedance, where the percentage varies according to the load areas. Reactive power loads are modeled as constant impedance throughout the system. Figure 6 depicts a pictorial view of the Brazilian Interconnections regions, highlighting the major interarea modes: the South/Southeastern (S/SE) mode, the North/ Northeastern (N/NE) mode and the North/Southern (N+NE/S+SE) mode.

Figure 6 – Interconnection sub-systems of BIPS The state-space representation of the open-loop system has 1516 state variables and the close-loop has 1638 state variables. The system matrices are obtained by the computer program PacDyn [9]. The objective is the tuning of 61 PSSs from the larger units of the BIPS. No simplifying assumptions regarding the actual PSS structures were made. The PSSs that remained out of the tuning process were left unchanged with respect to their current parameters. Results of small signal analysis The open-loop eigenvalues of the BIPS were obtained for three operating conditions using the PacDyn software [9]. Open-loop system The results are shown in Figure 7 Note that the BIPS is open-loop unstable.

Figure 7 – Open-loop eigenvalues for the BIPS Pole-zero mapping To verify solution feasibility a priori, we have computed the multivariable zeros for the 61-input 61-output transfer function. The result is shown in Figure 8 for the heavy-load operating condition. One can note that there are some pole-zero cancellations for some electromechanical modes. Taking as an example the cancellation around 11% damping ratio, modal analysis [15], [16] and [17] indicates that this mode corresponds to a local mode for the Cachoeira Dourada power plant, which is not equipped with PSS. So, beforehand, we know that the PGA methodology will not be able to shift this mode further into the left half complex plane.

-3 -2.5 -2 -1.5 -1 -0.5 0

0

2

4

6

8

10

12 0.2 0.15 0.1 0.05

Real Axis

Imag

inar

y A

xis

Pole-Zero Map

5%10%15%20%

Figure 8 – Multivariable pole-zero mapping. Figure 9 shows a zoomed region around Cachoeira Dourada local mode with some mode characteristics. Due to this kind of pole-zero cancellation it was considered a new fitness function that weights more the interarea electromechanical modes.

-1.5 -1 -0.56

7

8

9

10

11

12

0.15

0.1

Real Axis

Imag

inar

y A

xis

Pole-Zero Map

10%

15%

Figure 9 – Zoomed region around Cachoeira Dourada local mode (damping = 11.2%).

Close-loop system Figure 10 shows the close-loop eigenvalues after tuning the 61 PSSs. Note that the system becomes stable with minimum damping ratio greater than 10%, achieved in a local mode in the light loading condition.

Figure 10 – Close-loop system poles with 61 PSS’s tuning

by PGA. The eigenvalue -0.189+j1.481 has about 13.9% of damping in all three loading conditions. This mode is not the N/S interarea mode. Further investigations revealed that this mode is related to a speed regulator loop. The N/S interarea mode is the eigenvalue -0.571+j2.051 with 27.1% of damping. As shown in Figure 10 all intearea modes have a damping ratio greater than 20% in all loading conditions. Mode shape analysis Speed mode shape shows how the set of machines behave in the oscillation mode. An example to identify North-Southeastern interarea mode is showed in Figure 11.

Figure 11 – Mode shape for the mode 0.579+j2.063 (N/S

mode). The figure points out the North-Northeastern machines in first quadrant oscillating against South-Southeastern machines in the third quadrant. Results of non linear simulations The best PSS tuning solutions are also validated in nonlinear simulation tests with the BIPS facing large perturbations as single-phase fault applied on Ibiuna 500 kV AC-Bus. The fault is cleared after 80 ms by opening two Ibiuna – Bateias 500 kV transmission lines that connect South to Southeast regions. Figure 12, Figure 13 and Figure 14 show the rotor angle for all machines for the heavy, median and light load conditions, respectively. Many contingencies were applied for testing control performance. For all contingencies analyzed the designed PSSs had adequate performance.

Figure 12 – All Machine Angles for heavy load.

Figure 13 – All Machine Angles for median load.

Figure 14 – All Machine Angles for light load.

Pareto Front Figure 15 shows a picture of a Pareto Front combining the best solutions for the damping ratio of the interarea versus the local electromechanical modes. It becomes clear that small variation in one mode degrades the damping of the other mode when the solution is at the Pareto Front. The Figure 16 is a zoom of the region marked with a square in Figure 15.

Figure 15- Pareto front.

Figure 16- Zoom to show the best local and interarea damping.

This figure also shows how PGA works. At the beginning some unstable and low damping solutions appear. Along the searching process, minimum damping increases and the best solution is reached. As shown in Figure 9 the local mode -1.9+j9.71 with 11.2% damping, has a zero located at the same position. In Figure 16 this mode has 11.64% damping. This means that there is not possibility to shift this pole and then this is the limit condition for it. Future work A Hierarchical parallel GA (HPGA) [10], as shown in Figure 17, could be implemented to reduce CPU time, where np is the population size. This method allows performing one generation in about the same time spent to perform one QR plus the communication time among CPUs.

CPU1

CPU2

CPU3

CPU4

CPUeig np

CPUeig 2

CPUeig 1

CPUeig np

CPUeig 2

CPUeig 1

CPUeig np

CPUeig 2

CPUeig 1

CPUeig np

CPUeig 2

CPUeig 1

Perform QR

Perform PGA with migration

QR QR QR QR QR QR

QR QR QR QR QR QR

Perform QR

Figure 17 - Simple example of hierarchal parallelization.

Conclusions The results presented in this paper showed that the use of a PGA is possible for the tuning of 61 PSSs in a realistic power system model in three loading scenarios. The results were obtained in the Brazilian Interconnected Power System with very few approximations made in the synchronous machines control loops. The hardware utilized was capable to compute the eigenvalues of a 1500-state matrix in about 80 s using the QR algorithm. With 64 parallel CPUs, the problem of tuning the most import PSSs of the entire full-modeled BIPS in three operating conditions, became automatic with a very reasonable time for a off-line task. One of the great advantages of using the proposed tuning method is that after each run there will be many different solutions to the problem. It is possible that the best solution does not satisfy all the practical control design requirements. A fine search among the solutions may be required by an expert. However, in future developments, human expertise can be captured and readily implemented in a more elaborated fitness function. References [1] N. G. Hingorani, “Power Electronics in Electric Utilities: Role of

Power Electronics in Future Power Systems”, in Proceedings of IEEE, vol. 76, Apr. 1988.

[2] E. V. Larsen, J. J. Sanchez-Gasca, and J. H. Chow, “Concepts for Design of FACTS Controllers to Damp Power Swings”, IEEE Trans. on Power Systems, vol. 10, nº 2, pp. 948-956, May 1995.

[3] G. N. Taranto, J. H. Chow, and H. A. Othman, “Robust Decentralized Control Design for Damping Power System Oscillations”, in Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, FL, Dec. 1994, pp. 4080-4085.

[4] A. L. B. do Bomfim, G. N. Taranto and D. M. Falcão, “Simultaneous Tuning of Power System Damping Controllers Using Genetic Algorithms”, IEEE Trans. On Power Systems, Vol.15, Nº.1, pp. 163-169, February 2000.

[5] G. N. Taranto and D. M. Falcao, “Robust Decentralized Control Design using Genetic Algorithms in Power System Damping Control”, in IEE Proceedings on Generation, Transmission and Distribution, vol. 145, Jan. 1998, pp.1-6.

[6] J. H. Holland, Adaption in Natural and Artificial Systems, MIT Press/Bradford Books edition.

[7] D. E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning, Addison Wesley, Reading, MA, 1989.

[8] Electric System National Operator - ONS, stability data base; http://www.ons.org.br/avaliacao_condicao/casos_perturbacoes.aspx.

[9] ELETROBRAS CEPEL – PacDyn, ANAREDE and ANATEM user manuals.

[10] E. Cantú-Paz, “A Survey of Parallel Genetic Algorithms”, Calculateurs paralleles, reseaux et systems repartis, vol 10, 1998.

[11] M. Gen, R. Cheng, Genetic Algorithms and Engineering Design, John Wiley & Sons, INC.

[12] J-. W. Kim, S. W. Kim, P. G. Park and T. J. Park, “On The Similarities between Binary-Coded GA and Real-Coded GA in Wide Search Space”, IEEE 2002.

[13] LAPACK eigenvalue subroutines. [14] Center for Service in High Performance Computing - NACAD -

COPPE/UFRJ, http://www.nacad.ufrj.br/index.php. [15] R. T. Byerly, D. E. Sherman, and R. J. Bernnon, “Frequency

Domain Analysis of Low Frequency Oscillations in Large Electric Power Systems”, Report EPRI EL-726, 1978.

[16] N. Martins, L. T. G. Lima, H. J. C. P. Pinto, and N. J. P. Macedo, A State of the Art Computer Program for the Analysis and Control for Small-signal Stability of Large Scale AC/DC Power Systems”, in Proceedings of IERE Workshop on New Issues in Power System Simulation, France, Mar. 1992, pp. 11-19.

[17] N. Martins and L. T. G. Lima, “Determination of Suitable Locations for Power System Stabilizers and Static VAR Compensator for Damping Electromechanical Oscillations in Large Scale Power Systems”, IEEE Trans. on Power Systems, vol. 5, no.4, pp 1455-1469, Nov. 1990.

Appendix Genetic Algorithms Basic Concepts Genetic Algorithm (GA) [6] and [7] is a technique for searching the best solution of a problem using intensive combinatorial tentative solutions. They combine and select chromosomes by genetic operators as crossover, mutation, elitism and selection, to create a new generation of tentative solutions (newer chromosomes). It is based on Darwin’s theory where the best individual survive and spread it genes to the offspring. Chromosome Representation Chromosomes are represented by real numbers to reduce intensive combinatorial tentative solutions. The length of them is equal to the number of parameters to be found as shown in Figure 18, where n is the number of PSSs to be tuned.

1aK 2

aK anK 1

aα 2aα a

nα 1aω 2

aω anω

Figure 18 – Chromosomes representation.

Selection The Selection operator used is called tournament where a specified number of individuals is selected, and the best among those is chosen to crossover. Elitism The best individual from the present population is copied to the next population to preserve the best genes.

Uniform arithmetic crossover It gives the same chance to the best or to the bad offspring to combine. Suppose two chromosomes chosen randomly by the selection operator to crossover, and the crossover points selected by a mask, as shown in Figure 19.

1aK 2

aK anK 1

aα 2aα a

nα 1aω 2

aω anω

1bK 2

bK bnK 1

bα 2bα b

nα 1bω 2

bω bnω

Figure 19 – Select crossover points.

The result will be reached by the expressions:

1 1

2 2

(1 )

(1 )

a a bi ii

b b ai ii

β ββ λ λ

β ββ λ λ

= + −

= + −

Where: aiβ and b

iβ are the offspring result; aiβ the first

parent parameter; biβ the second parent parameter; 1λ

and 2λ are random values between 0.0 and 1.0. The crossover result is shown in Figure 20:

1aK 2

aK anK 1

aα 2aα a

nα 1aω 2

aω anω

1bK 2

bK bnK 1

bα 2bα b

nα 1bω 2

bω bnω

Figure 20 – Crossover result.

The Uniform Arithmetic Crossover shows a disruptive effect, since the factors λ1 and λ2 change the parameter value. In this way, new values are created in the new population. Exponential Variation Mutation Rate Mutation operator introduces new values in the population. The value is calculated by the following expression:

2 2*a anewα β α=

Where α 2a

new is the new value after mutation; β is a random number in the interval [0.0-1.0] and α 2

a is the random position value. As real number chromosomes representation is used, mutation operator is called uniform mutation, because the random numbers from the interval of each variable have the same rate to be chosen by mutation. Exponential variable mutation rate was used. After each generation, mutation rate is changed by the expression:

*exp( * )mut mutori gerP P def N= −

If then mut mutmin mut minP P P P⟨ = Where mutP is the exponential mutation rate; mutoriP the original mutation rate; mutminP the minimum mutation rate; gerN the number of generation for each search space and def is the decreasing exponential factor. Variable search space The PGA is customized to consider variable searching space and variable mutation rate. Reduction Space Bounds The values of new parameter bounds consider the best solution as reference.

Lower bound:

( ) ( )*(1 )( ) ( )*(1 )(2* ) (2* )*(1 )

lb best r

lb best r

lb best r

K m K m fnctrl m nctrl m f

nctrl m nctrl m fα αω ω

= −+ = + −

+ = + −

Upper bound:

( ) ( )*(1 )( ) ( )*(1 )(2* ) (2* )*(1 )

ub best r

ub best r

ub best r

K m K m fnctrl m nctrl m f

nctrl m nctrl m fα αω ω

= ++ = + +

+ = + +

Where 0.5rf < Expanded Space Bounds The expanded search space is created, first shifting the best solution by a factor sf .

( ) ( )*shiftbest sbest n n fX X=

Where bestX is the best solution; shift

bestX the solution shifted; n the chromosome length and sf >1.0 is the shifting factor. The expanded search space is created applying an expanding factor. Lower bound values are:

( ) ( )*(1 )

( ) ( )*(1 )

( ) ( )*(1 )

exp shiftelb best

exp shiftelb best

exp shiftelb best

K m K m f

m m f

m m f

α α

ω ω

= −

= −

= −

Upper bound values are:

( )*(1 )( )

( )*(1 )( )

( )*(1 )( )

shiftexpeub best

shiftexpeub best

shiftexpeub best

m fK m K

m fm

m fm

α α

ω ω

+=

+=

+=

Where 0.5ef >