binary real coded firefly algorithm for solving unit commitment problem

Information Sciences 249 (2013) 67–84

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Information Sciences

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Binary real coded firefly algorithm for solving unit commitmentproblem

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.06.022

⇑ Corresponding author. Tel.: +91 9944073421.E-mail addresses: [email protected] (K. Chandrasekaran), [email protected] (S.P. Simon), [email protected] (N.P. Padhy).

K. Chandrasekaran a, Sishaj P. Simon b,⇑, Narayana Prasad Padhy c

a Department of EEE, Kamaraj College of Engineering & Technology, Virudhunagar 620001, Indiab Department of EEE, National Institute of Technology, Tiruchirappalli 620015, Indiac Department of EE, Indian Institute of Technology, Roorkee 247667, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 March 2012Received in revised form 30 April 2013Accepted 8 June 2013Available online 15 June 2013

Keywords:Binary real coded firefly algorithmEconomic dispatch problemUnit commitment problem

This paper presents a new biologically-inspired binary real coded firefly (BRCFF) algorithmto solve the unit commitment problem (UCP) by considering system and generating unitconstraints. The firefly (FF) algorithm is inspired by the flashing behavior of fireflies andthe phenomenon of bioluminescent communication. Upon solving UCP, the proposed bin-ary coded FF algorithm determines the ON/OFF status of the generating units, while theeconomic dispatch problem (EDP) is solved using the real coded FF algorithm. The mannerof firefly communication through luminescent flashes and their synchronization is imitatedand suitably implemented in UCP. An effective constraint handling mechanism is intro-duced to solve complicated system and unit constraints. Finally, the proposed algorithmis applied to 3, 12, 17, 26, and 38 generating unit systems for a 24 h scheduling horizonand a comparative study is conducted using other recently reported results. Numericalresults clarify and verify the significance of the proposed algorithm. The results obtainedindicate that the proposed biologically-inspired algorithm could be an important playerin swarm-based optimization.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The optimum economic operation and planning of electric power generation systems occupies a crucial position in theelectric power industry. Unit commitment is an important function in the generation of resource management in a powersystem. A unit commitment problem (UCP) in a power system is a combinatorial optimization problem that determinesthe ON/OFF status of the generating unit in order to satisfy the load demand, spinning reserve, and physical and operationalconstraints of the individual unit. The economic dispatch problem (EDP) [57,58] (is the sub problem of UCP) determines theoptimal dispatch among the committed generating units during each period of operation in order to satisfy the system load.Therefore, UCP is a nonlinear mixed integer programming problem and is a computationally expensive proposition for largepower systems.

1.1. Literature survey for UCP

In UCP, power system operators have to maintain a certain amount of generation capacity as spinning reserve [8–10]. Thisapproach ensures that the power system is able to withstand the sudden outage of some generating units, transmission lines,or an unforeseen increase in load, without having to resort to load shedding. In the deterministic method of setting spinning

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Nomenclature

Fc (Pi,k) fuel cost ($)ai, bi, ci cost coefficient of ith generator unitH total number of hours consideredIi,k status of unit i at kth hour. (i.e.) 1 for ON and 0 for OFFLoadk total system demand at kth hourLOLPspec

k; specified LOLP at kth hourLOLPk calculated LOLP at kth hourPi,k generation power output of unit i at kth hourPi,max maximum power output of unit iPi,min minimum power output of unit ihci hot start costcci cold start costRU(i), RD(i) ramp up/ramp down rate limit of unit iSCi,k startup cost of unit i at kth hourSRk system spinning reserve in MW at kth hourTon(i), Toff(i) minimum ON/OFF time for unit iXon(i,k) time duration for which unit i is ON at kth hourXoff(i,k) time duration for which unit i is OFF at kth hourTOC total operating costc absorption coefficientr distance between two firefliesn constant

Indicesi, k generating unit and time index, respectivelyj jth state of capacity outage probability tablep, q firefly position

68 K. Chandrasekaran et al. / Information Sciences 249 (2013) 67–84

reserve, the minimum amount of spinning reserve is set to a value that is at least equal to the capacity of the largest unit, orto a specific percentage of the hourly system load [12–14,35,37–39].

1.2. Literature survey for solution techniques

Many optimization techniques are available for solving the nonlinear mixed integer programming problem [59]. A liter-ature review of the UCP and the solution techniques are given in [32,42]. The classical optimization methods such as the pri-ority list method [2], dynamic programming [46], branch and bound [4,6], mixed integer programming [23,28], andLagrangian Relaxation (LR) [29,60] are widely used conventional techniques. The priority list method is simple and fast.However, it produces a sub-optimal solution with a higher operation cost. The dynamic programming method has dimen-sionality problems, whereby the execution time increases rapidly with the number of generating units to be committedas the size of the problem increases. The LR method provides a quick solution. However, the dual nature of the algorithmsuffers from numerical convergence and solution quality. In the branch-and-bound and mixed integer programming meth-ods, the computational time increases substantially for a large-scale power system. Therefore, artificial intelligence tech-niques such as neural networks [31], expert systems [51], genetic algorithms (GA) [7,19], simulated annealing (SA) [24],evolutionary programming (EP) [18], tabu search [25], fuzzy logic [40], particle swarm optimization (PSO) [16,36,41,48],ant colony optimization (ACO)[5,43], frog leaping algorithm (FLA) [17], tuned harmony search algorithm (HAS) [33], teach-ing–learning algorithm [26], T-cell algorithm [49] and migrating birds optimization (MBO) [11] are often used. These solu-tion techniques are good for searching the near global optimal solution and can be considered successful to a certain extent.Since new swarm-based optimization techniques are also emerging, the best commitment solution with less computationaltime is a challenging task within the research community. In recent years, a new biologically-inspired meta-heuristic algo-rithm, known as the firefly algorithm and developed by Xin-She Yang, has been successfully used to solve the nonlinear andnon-convex optimization problems [15,53–56]. Based on these previous findings, we attempted to demonstrate a new meth-odology for solving the UCP problem with the FF algorithm, with the goal of providing a practical alternative for conventionalsolution methods.

K. Chandrasekaran et al. / Information Sciences 249 (2013) 67–84 69

2. Proposed work

The FF algorithm, developed by Yang [53], is used to solve various mixed variables and constrained engineering optimi-zation problems [3,27,47,53–56] that are found to be successful. In Refs. [53–56], the FF algorithm is tested and validated forsolving the power system economic dispatch problem and is found to be successful. A more detailed description on the the-ory of the FF algorithm is provided in Refs. [3,54]. In this paper, we attempt to solve the unit commitment problem by con-sidering system and generating unit constraints using a binary real coded firefly (BRCFF) algorithm.

This paper proposes the implementation of the BRCFF algorithm to solve the unit commitment problem. Here, a novelbinary coded FF algorithm is proposed to solve the UCP and the real coded FF algorithm is used to solve the EDP. A tanh func-tion is introduced in the binary coded FF algorithm and an improvement in the performance is made with respect to othermethods available in the literature (given in the Section 7.2).

The rest of the sections in this paper are organized as follows: In Section 3, the problem formulation of UCP is presentedwith system and generating unit constraints. Sections 4 and 5 explain the basic behavior of FF and its implementation for thebinary-coded FF algorithm for solving UCP. Section 6 explains the solution methodology of the proposed FF algorithm forsolving UCP. In Section 7, the effectiveness of the proposed methodology is demonstrated on different test systems andthe results are discussed. Finally, a conclusion is provided in Section 8.

3. Problem formulation of UCP

The objective of UCP is to determine the ON/OFF status of the generating unit such that the total operating cost is min-imized subject to the system and generating unit constraints. Mathematically, the objective of UCP is to minimize the totaloperating cost (TOC), which includes fuel cost and start-up cost.

The fuel cost of a thermal generating unit is expressed as a second order function of each of the unit’s output as:

FcðPi;kÞ ¼ ai þ bi:Pi;k þ ci:P2i;k ð1Þ

The start-up cost of a generating unit depends on the period of time during which the unit is previously off and whetherthe boilers are kept hot during the shut-down period. Thus, the start-up cost can be defined by

SCi;k ¼hci when Toff

i 6 Xoff ði; kÞ 6 Toffi þ cshi

cci when Xoff ði; kÞ > Toffi þ cshi

( )ð2Þ

The overall objective function of the UCP is given in

Min TOC ¼XH

k¼1

XN

i¼1

½FcðPi;kÞ � Ii;k þ SCi;k� ð3Þ

which is subject to the constraints (4)–(10).The variable Ii,k in Eq. (3) indicates the status (i.e. 1 for ON and 0 for OFF) of the generating units, which is determined in

the UCP using the binary-coded FF algorithm. The variable Pi,k in Eq. (3) indicates the output power of the generating unit,which is determined in the EDP using a real coded FF algorithm.

3.1. System constraints

3.1.1. Power balance constraintThe total generated power at each hour must be equal to the load of the corresponding hour. This constraint is given be-

low in

XN

i¼1

ðP�i;kIi;kÞ ¼ Loadk; k 2 ½1;H� ð4Þ

3.1.2. Spinning reserve constraintThe spinning reserve is the total amount of real power generation available from all of the synchronized units minus the

present load. The reserve is considered to be a pre-specified amount or a given percentage of the foretold peak demand [1].Thus, it should be sufficient enough to meet the loss of the most heavily loaded unit in the system.

This constraint is given below in

XN

i¼1

ðP�i;maxIi;kÞP Loadk þ SRk; k 2 ½1;H� ð5Þ


3.2. Generating unit constraints

3.2.1. Generator capacity limitEach unit has a generation limit and is given in

Pi;min 6 Pi;k 6 Pi;max ð6Þ

3.2.2. Unit minimum ON/OFF durationThe total number of hours, for which unit i has been running (Xon), must be greater than or equal to the minimum unit up-

time (Ton). This constraint is given below in

Xonði; kÞP TonðiÞ ð7Þ

The total number of hours, for which unit i has been down (Xoff), must be greater than or equal to the minimum unitdown-time (Toff). This constraint is given below in

Xoff ði; k� 1ÞP Toff ðiÞ ð8Þ

3.2.3. Unit ramp constraintsDue to the physical restrictions on thermal generating units, the rate of change in generation must be limited within a

certain range. The limits of the ramp rate confines the change of output of a generating unit between adjacent hours.For each unit, the output is limited by the ramp up/down rate at every hour as given below:

Pi;k � Pi;k�1 6 RUðiÞ as generation increases ð9ÞPi;k�1 � Pi;k 6 RDðiÞ as generation decreases ð10Þ

4. Overview of FF algorithm

Nature-inspired methodologies are among the most powerful algorithms for optimization problems. The FF algorithm is anovel nature-inspired algorithm inspired by the social behavior of fireflies. By idealizing some of the flashing characteristicsof fireflies, a firefly-inspired algorithm was presented by Xin-She Yang. The pseudo code of the firefly-inspired algorithm wasdeveloped using these three idealized rules:

� All fireflies are unisex and are attracted to other fireflies regardless of their sex.� The degree of the attractiveness of a firefly is proportional to its brightness, and thus for any two flashing fireflies,

the one that is less bright will move towards the brighter one. More brightness means less distance between twofireflies. However, if any two flashing fireflies have the same brightness, then they move randomly.

� Finally, the brightness of a firefly is determined by the value of the objective function. For a maximization problem,the brightness of each firefly is proportional to the value of the objective function and vice versa.

4.1. Degree of the attractiveness of a firefly

In the FF algorithm, the main form of attractiveness function b can be any of the monotonically decreasing functions asgiven in

bðrÞ ¼ b0e�crnpq ; n P 1 ð11Þ

The attractiveness can be achieved by tuning the parameters b0 and c. The distance rpq between two fireflies is given in

rpq ¼ kVp � Vqk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXd

s¼1ðVp;s � Vq;sÞ2

rð12Þ

where Vp,s is the sth component of the spatial coordinate of the pth firefly and d is the number of dimensions. Theq 2 {1,2, . . . ,m} is a randomly chosen index. Although q is determined randomly, it has to be different from p. Here, m isthe number of fireflies. For other applications such as scheduling, the distance can be any of the suitable forms, and not nec-essarily the Cartesian distance (Eq. (12)). In general, b0 2 [0,1], and this when b0 = 0, only a non-cooperative distributed ran-dom search is applied. When b0 = 1, the scheme of a cooperative local search is performed such that the brightest fireflystrongly determines the other fireflies’ position, especially in its neighborhood [53,54]. The value of c determines the vari-ation of attractiveness, which corresponds to the variation of distance from the communicated firefly. When c = 0, there is novariation or the fireflies have constant attractiveness. When c =1, it results in attractiveness being close to zero, which againis equivalent to the complete random search. In general, the value of c[53,54] is in between [0,10].


4.2. Modification of firefly position

The movement of a firefly p, when attracted to another more attractive (brighter) firefly q, is determined by

V 0p ¼ Vp þ bðrÞ � ðVp � VqÞ þ a rand� 12

� �ð13Þ

where V 0pq is the firefly position of the next generation. Vp and Vq are the current position of the fireflies and V 0p is the pthfirefly position of the next generation. The second term in Eq. (13) is due to attraction. The third term introduces random-ization, with ‘a’ being the randomization parameter and ‘‘rand’’ is a random number generated uniformly but distributedbetween 0 and 1.

5. Implementation of binary coded FF algorithm for solving UCP

In UCP, binary numbers 0 and 1 are used to indicate the unit status (i.e., OFF or ON). The firefly algorithm used in [53–56]is essentially a real-coded algorithm, and therefore some modifications are needed to enable it to deal with the binary var-iable (i.e., 0 and 1) optimization problem. In the proposed binary coded FF algorithm, Vp and Vq in Eq. (13) is represented by Ip

and Iq. It takes a value of 0 or 1 to denote the OFF or ON status of the generating units, respectively. The third term is therandom number generated in the range (�1,1). When b0 = 1 and c = 1, the b value in Eq. (11) takes values between0.3679 and 2.7183. However, the corresponding variation of the rpq value varies between �1 to 1, which can be calculatedusing Eq. (14).

rpq ¼ Ip � Iq ð14Þ

When the position of the firefly is modified, I0p will vary between �2.2180 to 4.3 and can be calculated using

I0p ¼ Ip þ bðrÞ � ðIp � IqÞ þ a rand� 12

� �ð15Þ

To determine the unit status as 1 or 0, a threshold level has to be fixed. If Ip is greater than the threshold, then the indi-vidual is more likely to choose ‘1’. If Ip is smaller than the threshold, then the individual is more likely to choose ‘0’. Thethreshold level can be made to range from 0 to 1, and in order to achieve this a sigmoid function [20,22,34] is often used,as given in

f I0p� �

¼ 1

1þ exp �I0p� � ð16Þ

However, to improve the performance of binary coded firefly algorithm, another function called tanh is used as given in

f I0p� �

¼ tanh jI0pj� �

¼exp 2� I0p

�� 1

exp 2� I0p��

þ 1ð17Þ

Both of these functions scale the Ipq value in the [0,1] range as shown in Fig. 1. A random number is generated between 0and 1 to decide the unit status as 0 or 1. If f I0p

� �is greater than rand (0,1), then the unit status is 1: otherwise it is 0. Based on

20 trial runs, it is observed that the performance of tanh function is faster than the sigmoid function on reaching a qualitysolution (see Section 7.1).

00.10.20.30.40.50.60.70.80.9

1

-2.2 -1.2 -0.2 0.8 1.8 2.8 3.8Ip'

f(Ip'

)

tanh functionSigmoid function

Fig. 1. Variation of Ip and f(Ip).

0110110101

0100100110

0.9238

0.0093

4.1681

0.6346

0.0997

2.8872

-0.2004

0.5197

0.2322

3.5811

0.7158

0.5023

0.9848

0.6535

0.5249

0.9472

0.4501

0.6271

0.5578

0.9729

0.7277

0.0093

0.9995

0.5612

0.0994

0.9938

-0.1978

0.4775

0.2281

0.9985

Randomly generated binary bits

Sigmoidfunction

Tanhfunction

(a) (b)

pth qth Ip'

0.9106

0.1818

0.2638

0.1455

0.1361

0.8693

0.5797

0.5499

0.1450

0.8530

0111110111

0011010011

Random numbergenerated between

0 and 1 to decide theunit status as 0 or 1

(f)(c) (d) (e) (g) (h)

Real coded

Modified Binary coded firefly position of pth firefly Ip'

Sigmoidfunction

Tanhfunction

Fig. 2. Binary-coded FF algorithm using sigmoid and tanh functions.


The modification of the firefly position (bits) with respect to sigmoid function and tanh function is shown in Fig. 2. Fig. 2ais the randomly generated bits of pth firefly position. Fig. 2b is the randomly generated bits of qth firefly position. The move-ment of a firefly p, when attracted to another more attractive (brighter) firefly q, is evaluated by (15) and is given in Fig. 2c

when b0 = 1 and c = 1. The f I0p� �

using sigmoid and tanh function is given in Fig. 2d and e, respectively. Fig. 2f is the random

number generated between 0 and 1 in order to decide the generating unit status as 0 or 1. The flipping of bits is carried outusing a sigmoid function, which is shown as light shading in Fig. 2g. The bit flipped using the tanh function is shown as darkshading in Fig. 2h. It is clear from Fig. 2g and h that the chances of bit flipping are greater in the case of the tanh function thanthe sigmoid function when compared with the initially generated pth firefly position. These results indicate that the pro-posed tanh function could be an important player in the proposed binary coded FF algorithm.

6. Solution methodology

In this section, the novel binary-coded FF algorithm using the tanh function is proposed to solve the UCP and the realcoded FF algorithm is used to solve the sub-problem EDP.

6.1. Step-by-step procedure for binary real coded FF algorithm for UCP

Step 1: Initialize the FF parameters (c,b0, size for the firefly population and the maximum number of generation Genmax, forthe termination process).

Step 2: Initialization of firefly positionInitialize randomly M = [X1; X2; X3; . . . ; Xm] of m solutions or firefly positions in the multi-dimensional search spacewhere m represents the size of the firefly population. Each solution of X is represented by the D-dimensional vector.Here, in UCP, D is equal to N⁄H.

Step 3: Modification of firefly positionA firefly produces a modification in the position based on the brightness between the fireflies. The new position isdetermined by modifying the value (old firefly position) using Eq. (15).

Step 4: Repair strategy for constraint managementWhenever the firefly position is modified, check for the constraints 5 and 7 and 8. If there is any violation in con-straint, then a repair strategy given in Section 6.2 is performed to overcome the violation. Otherwise, go to step 5.

Step 5: Solve EDPOnce the violation is overcome, EDP is then solved for each feasible position and, constraints 4, 6 and 9, 10 are sat-isfied. The real-coded FF algorithm as mentioned in Section 6.3 is used to solve the EDP.

Step 6: Evaluation of fitness of the populationEvaluate the fitness value of each firefly position corresponding to the brightness of the firefly. The fitness value isevaluated using

FITp ¼ 1=Costp; if Costp > 0¼ 1þ absðCostpÞ; if Costp < 0

ð18Þ

Costp is the cost of generation of pth firefly position. Cost is calculated from the EDP. In (18) the individual with the lowestcost has the highest fitness. Set the generation count as one and repeat the following steps until the maximum generationnumber (Genmax) of the termination criteria is reached.


Step 7: Memorize the best solution achieved so far. Increment the generation count.Step 8: Stop the process and display the result if the termination criteria are satisfied. Termination criteria used in this work

are the specified maximum number of generations. Otherwise, go to step 3.

6.2. Repair strategy in binary-coded FF algorithm

Whenever the commitment status of the generating unit for each time interval is generated randomly or if the firefly po-sition is modified, the violation of minimum up/down time constraints (7) and (8) and the spinning reserve constraint (5)have to be checked as follows.

Step 1: If the spinning reserve (5) is met, then go to step 3. Otherwise, go to the next step.Step 2: The less expensive units, which are in the OFF state, are identified and turned ON. Then, go to step 1.Step 3: If the spinning reserve constraint is satisfied, then the minimum up and down time constraints (7) and (8) are

checked for each unit. If there is any violation in the minimum up or down time constraint then a repair schemeis performed to overcome the violation. For instance, let us assume that the Toff for a hypothetical unit is 4. For ascheduling interval of 12 h, if the actual off time for that unit is 3 h (4th–6th hour), then it violates the Toff constraint.In this case, the unit status before the 3rd hour or after the 7th hour should be made OFF to mitigate the violation. Bymaking this change, if it violates the Ton constraint, then the status of the unit is made ON.

Step 4: The repair scheme in step 3 may affect the spinning reserve constraint of the system. If the spinning reserve con-straint is met, then accept the feasible solution. Otherwise, go to step 2.

A minimum number of trials should be set for the repair mechanism. These steps are carried out for the entire hourlyload.

6.3. Real-coded FF to solve EDP

6.3.1. Initialization of real string population (real coded FF algorithm)Generate a random population of R initial solutions represented by real values for the pth string of the population with S

dimensional vector. It should be noted that the real values will be initialized only for the generating unit that has an ON sta-tus obtained by the binary-coded FF algorithm. Hence, randomly initialize a population R = [Y1; Y2; Y3; . . . ; Ym]H of m solu-tions or firefly positions in the multi-dimensional search space, where m represents the firefly population. Each solution of Yis represented by the S-dimensional vector. Here, in EDP, S is equal to Non. Non is the total number of ON generating units.

6.3.2. Repair strategy for constraint management in real coded FFWhenever the firefly position is modified, it should satisfy the constraints, as given in Eqs. 6, 9 and 10. If there is any vio-

lation in the constraint, then a repair strategy is performed to overcome the violation. The steps given below are used toovercome the violations.

Step 1: If the generated power (Pi) capacity limit (6) is met, then go to step 3. Otherwise, go to next stepStep 2 : If Pi > Pi,max, then Pi = Pi,max. If Pi<Pi,min, then Pi = Pi,min and go to next step.

Step 3: Check if power shortage occurs at kth hour due to start up and shut down ramp rate constraints. In case of any violation,go to step 4, otherwise, go to step 5.

Step 4: Extend the commitment of unit i + 1 at hour k to increase power at this hour.Step 5: End

6.3.3. Pseudo code of real coded FF to solve EDP

1. Initialize the control parameters and read the system data.2. Set the time interval, k = 1.3. Generate the initial population as in Section 6.3.1.4. Set the generation count, Gen = 1.5. For each firefly position produce a new solution using (13).6. Check for constraints (6). If the constraint (6) is violated, then perform repair strategies as given in Section 6.3.2, then cal-

culate the fitness value using (18).7. Memorize the best solution achieved so far.8. Gen = Gen + 1.9. Until Gen = maximum number of generation (Genmax), go to step 5.

10. k = k + 1.11. Until k = 24, go to step 3.12. End.


7. Result and discussions

The proposed BRCFF algorithm is tested on 3, 12, 17, 26, and 38 unit systems. In all cases, the ON/OFF status of the gen-erating units is obtained using the binary-coded FF algorithm and the real coded FF algorithm is used to solve the EDP.

7.1. Parameter settings

Good convergence behavior can be obtained if the four control parameters, namely b0, c, n, and firefly population size, canbe optimally tuned. The optimal tuning of these firefly parameters would also yield a better quality solution with less com-putational time. By default, one of the parameters is initially varied and the other parameters are kept constant. Each param-eter as tested by taking into account several values within a boundary range discussed in the Section 4.1. Ten simulations foreach setting are performed in order to achieve some statistical information about the average evolution. Based on the aboveguidelines, numerical analysis is carried out to obtain the best selection of parameter values. In the binary-coded FF algo-rithm, the value of n in Eq. (11) is always set as 1and the firefly population size is varied between 20 and 100. Similarly,in the real coded FF, the value of n in Eq. (11) is varied between 1 and 3 and the firefly population size is varied between25 and 200. For all of the trial runs, the maximum iteration limit is set to 300 in the binary-coded FF algorithm.

The control parameters are selected based on the average total operating cost (TOC) obtained from 10 simulations. Out of10 simulations, the minimum value of average TOC is shown as the shaded portion in Table 1. Hence, the final combination ofparameters for the binary-coded FF algorithm that provided the best results is c = 0.5, b0 = 0.4, and a firefly population size of20. Similarly, the real coded FF algorithm parameter tuning was conducted and the final combination of parameters that pro-vided the best result was c = 0.4, b0 = 0.6, n = 1 and a firefly population size of 100. It should be noted that once the param-eters are finalized for the system, the same settings can be used to obtain the best solution even if there is a change in theforecasted load.

Similarly, for 12, 17, 26 and 38 unit systems, the best combination of control parameters are evaluated and are given inTable 2.

7.2. Test system 1:3 unit system

A smaller test system comprised of 3 generating units [13] is considered. The generating unit data and load profile for aperiod of 24 h are adapted from the same reference, and the spinning reserve at each hour is considered as in the ref [13] (i.e.,5% of the daily peak load).

Out of 30 trial runs, the best TOC obtained for the 3 unit system for a 24-h time interval is $178611.57. The detail statusand power dispatch of generating units for a 24-h time interval are given in Table 3. Table 4 provides a comparison of theTOC obtained from the BRCFF algorithm with that of the genetic algorithm using a specialized operator approach available inthe literature. The minimum cost to date reported in the literature is $179116.00 [13], which is $504.43 higher than thatobtained from the BRCFF. Thus, the proposed method yields better results and show the superiority of the FF algorithm.

It is inferred from Fig. 3 that the characteristics of the binary FF algorithm using tanh function steadily reaches the min-imum value after a few iterations compared to the sigmoid function. It can be concluded that the convergence characteristicof a binary FF using the tanh function is faster than the binary FF that uses a sigmoid function. Hence, the binary FF using thetanh function produces a quality solution in less time compared to other techniques.


A test system comprised of 12 generating units [13] is considered. The generating unit data and load profile for a period of24 h are adapted from the same reference, and the spinning reserve at each hour is considered as in the ref [13] (i.e., 5% of thedaily peak load).

Table 1Parameter settings of a binary-coded FF-3 unit system.

c Average total operating cost $ b0 Average total operating cost $ Firefly population size Average total operating cost $

0 178,913 0 179,432 10 178,9520.5 178,821 0.1 179,014 20 178,7031 179,216 0.2 179,148 30 178,7351.5 179,263 0.3 178,947 50 178,8922 179,387 0.4 178,721 100 178,9465 179,523 0.5 179,41710 179,314 0.6 179,143

0.7 178,9760.8 178,8860.9 179,2261 179,279

Table 2Parameter settings of the BRCFF algorithm.

Parameter 12 unit system 17 unit system 26 unit system 38 unit system

Binary FF Real FF Binary FF Real FF Binary FF Real FF Binary FF Real FF

c 0.5 1 0.5 0.5 1 0.9 1.5 2b0 0.6 0.4 0.6 0.9 0.8 0.9 0.4 0.9n 1 2 1 1.5 1 1 1 1.5Firefly population size 20 100 50 100 100 100 100 100Max. Iter. 300 500 300 500 300 500 300 500

Table 3UCP status and power dispatch-3 unit system.

Hour Unit no. Fuel cost $ Startup cost $ Total operating cost $

1 2 3

1 252.74 0 234.76 5489.60 0 5489.602 238.73 0 221.27 5225.00 0 5225.003 239.24 0 221.76 5234.50 0 5234.504 233.63 0 216.37 5129.60 0 5129.605 235.80 0 218.45 5170.10 0 5170.106 243.83 0 226.17 5320.80 0 5320.807 253.00 0 235.00 5494.40 0 5494.408 317.08 0 296.67 6749.40 0 6749.409 236.94 211.51 219.55 7666.80 5330.24 12997.00

10 237.54 212.09 220.12 7683.50 0 7683.5011 232.33 207.06 215.11 7538.30 0 7538.3012 244.70 219.03 227.02 7884.80 0 7884.8013 250.85 224.97 232.93 8058.30 0 8058.3014 250.85 224.97 232.93 8058.30 0 8058.3015 222.95 197.98 206.08 7278.10 0 7278.1016 234.04 208.71 216.75 7585.80 0 7585.8017 283.44 256.51 264.30 8995.90 0 8995.9018 307.59 279.87 287.54 9708.70 0 9708.7019 292.66 265.42 273.17 9266.10 0 9266.1020 289.93 262.78 270.54 9185.80 0 9185.8021 277.38 250.65 258.47 8819.50 0 8819.5022 253.66 227.70 235.64 8138.20 0 8138.2023 215.01 190.30 198.44 7060.00 0 7060.0024 187.50 180.00 180.00 6539.90 0 6539.90

Total ($) 173281.40 5330.24 178611.57

Table 4Comparison of results-3 unit system.

Solution technique Minimum operating cost ($)

Genetic algorithm with specialized search operators [13] 179116.00BRCFF algorithm 178611.57


Out of 30 trials, the best TOC obtained for the 12 unit system for a 24-h-time interval is $639938.60. The detailed statusand power dispatch of generating units for a 24-h time interval are given in Table 5. Table 6 provides a comparison of theTOC obtained from the BRCFF algorithm with that of other techniques. The minimum TOC to date reported in the literature is$644951.00 [13], which is $5012.40 higher than that obtained from BRCFF.


A test system comprised of 17 generating units [10] is considered. The generating unit data and load profile for a period of24 h are adapted from the same reference, and the spinning reserve at each hour is set as 100 MW as in the ref [10].

Out of 30 trials, the best TOC obtained for the 17 unit system for a 24-h-time interval is $1014390. The detailed status andpower dispatch of generating units for a 24-h time interval are given in Table 7. Table 8 provides a comparison of the TOCobtained from the BRCFF algorithm with that of other techniques. The minimum TOC to date reported in the literature is$1016062.00 [8], which is $1672 higher than that obtained from BRCFF.

178500178600178700178800178900179000179100179200179300179400179500179600

0 50 100 150 200 250 300Iteration no.

Tota

l ope

ratin

g co

st ($

)

BRCFF-tanh funBRCFF-sigmoid fun

Fig. 3. Convergence graph using tanh and sigmoid function.


Hour Unit no. Fuelcost $

Startupcost $

Totaloperatingcost $

1 2 3 4 5 6 7 8 9 10 11 12

1 0 0 0 180.00 180.00 180.00 180.00 282.06 0 350 290.26 307.68 19,839 0 19839.002 0 0 0 180.00 180.00 180.00 180.00 253.6 0 323.95 262.47 279.99 18,896 0 18896.003 0 0 0 180.00 180.00 180.00 180.00 254.6 0 324.98 263.45 280.97 18,930 0 18930.004 0 0 0 180.00 180.00 180.00 180.00 243.53 0 313.63 252.64 270.2 18,559 0 18559.005 0 0 0 180.00 180.00 180.00 180.00 247.81 0 318.01 256.81 274.36 18,702 0 18702.006 0 0 0 180.00 180.00 180.00 180.00 263.66 0 334.26 272.29 289.78 19,236 0 19236.007 0 0 0 180.00 180.00 180.00 180.00 282.74 0 350.00 290.92 308.34 19,856 0 19856.008 0 0 0 234.77 292.6 267.94 259.69 350.00 0 350.00 350.00 350.00 24,490 0 24490.009 0 0 238.08 229.94 287.01 262.8 254.17 350.00 0 350.00 350.00 350.00 27,004 5321.10 32325.1010 0 0 239.42 231.24 288.51 264.18 255.65 350.00 0 350.00 350.00 350.00 27,073 0 27073.0011 0 0 227.72 219.95 275.44 252.15 242.74 350.00 0 350.00 350.00 350.00 26,482 0 26482.0012 0 0 255.54 246.79 306.52 280.73 273.43 350.00 0 350.00 350.00 350.00 27,896 0 27896.0013 0 0 269.35 260.11 321.95 294.92 288.67 350.00 0 350.00 350.00 350.00 28,613 0 28613.0014 0 0 269.35 260.11 321.95 294.92 288.67 350.00 0 350.00 350.00 350.00 28,613 0 28613.0015 0 0 206.62 199.59 251.87 230.47 219.45 350.00 0 350.00 350.00 350.00 25,435 0 25435.0016 0 0 231.56 223.65 279.73 256.09 246.97 350.00 0 350.00 350.00 350.00 26,675 0 26675.0017 0 281.05 288.71 278.79 343.58 314.82 310.03 350.00 0 350.00 350.00 350.00 32,794 5330.20 38124.2018 0 284.45 292.1 282.06 347.36 318.3 313.76 350.00 261.97 350.00 350.00 350.00 36,038 5328.30 41366.3019 0 260.02 267.79 258.61 320.21 293.33 286.95 350.00 238.08 350.00 350.00 350.00 34,268 0 34268.0020 0 255.55 263.35 254.33 315.25 288.76 282.05 350.00 233.71 350.00 350.00 350.00 33,949 0 33949.0021 0 235.03 242.94 234.63 292.44 267.79 259.53 350.00 213.64 350.00 350.00 350.00 32,500 0 32500.0022 0 195.52 203.64 196.72 248.54 227.41 216.17 350.00 180 350.00 350.00 350.00 29,836 0 29836.0023 0 180.00 180.00 180.00 190.18 180.00 180.00 314.2 0 350.00 321.65 338.96 25,113 0 25113.0024 0 180.00 180.00 180.00 180.00 180.00 180.00 251.08 0 321.37 260.01 277.54 23,162 0 23162.00

Total ($) 623,959 15979.60 639938.60



Simulated annealing [13] 702379.00Heuristic method [13] 665634.00Genetic algorithm with specialized search operators [13] 644951.00BRCFF algorithm 639938.60



A test system comprised of 26 generating units [44,52] is considered. The fuel cost coefficient and load profile is adaptedfrom the same reference. The system spinning reserve is set to the maximum capacity of the largest committed unit as in Ref.[52] and results are compared to [30,50,52]. The control parameters are selected for 26 units based on the average TOC (Ta-ble 2). The generating unit status and hourly operating cost of generating units are given in Table 9. Out of 30 trials, the best


Hour Unit no. Fuel cost $ Startup cost $ Total operating cost $

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 330 298 154 123 234 246.00 0 0 161.00 276.00 0 0 114 126 0 0 0 36,004 5528 41,5322 330 298 154 123 234 246.00 0 0 54.00 265.00 0 0 55 64 0 0 0 30,757 0 30,7573 330 298 154 123 234 243.9 0 0 54.00 155.1 0 0 55 64 0 0 0 28,621 0 28,6214 330 298 154 123 234 205.3 0 0 54.00 113.7 0 0 55 64 0 0 0 27,152 0 27,1525 330 298 154 123 234 198.5 0 0 54.00 106.5 0 0 55 64 0 0 0 26,898 0 26,8986 330 298 154 123 234 198.00 0 0 54.00 106.00 0 0 55 64 0 0 0 26,880 0 26,8807 330 298 154 123 234 237.6 0 0 54.00 148.4 0 0 55 64 0 0 0 28,381 0 28,3818 330 298 154 123 234 246.00 0 0 167.00 276.00 0 0 114 126 0 0 0 36,171 0 36,1719 330 298 154 123 234 246.00 91 0 164.00 276.00 0 0 114 126 0 118.00 62.00 43,110 4667 47,77710 330 298 154 123 234 246.00 91 0 172.00 276.00 0 159.00 114 126 0 118.00 62.00 47,689 2650 50,33911 330 298 154 123 234 246.00 91 0 168.00 276.00 0 159.00 114 126 0 118.00 62.00 47,576 0 47,57612 330 298 154 123 234 246.00 91 0 127.4 276.00 0 159.00 114 126 0 113.6 62.00 46,382 0 46,38213 330 298 154 123 234 246.00 91 0 157.00 276.00 0 159.00 114 126 100 118.00 62.00 50,150 2870 53,02014 330 298 154 123 234 246.00 91 0 94.22 276.00 0 115.8 114 126 100 52.00 28.00 44,999 0 44,99915 330 298 154 123 234 246.00 91 0 125.6 276.00 0 159.00 114 126 100 89.42 28.00 47,742 0 47,74216 330 298 154 123 234 246.00 91 0 125.4 276.00 0 159.00 114 126 100 87.56 28.00 47,692 0 47,69217 330 298 154 123 234 246.00 91 0 126.00 276.00 0 159.00 114 126 100 95.06 41.9 48,247 0 48,24718 330 298 154 123 234 246.00 91 0 162.00 276.00 0 159.00 114 126 100 118.00 62.00 50,287 0 50,28719 330 298 154 123 234 246.00 91 95 173.00 276.00 0 159.00 114 126 100 118.00 62.00 53,057 3789 56,84620 330 298 154 123 234 246.00 91 95 126.4 276.00 0 159.00 114 126 100 100.9 60.6 51,342 0 51,34221 330 298 154 123 234 246.00 91 95 127.1 276.00 0 159.00 114 126 0 109.00 62.00 48,718 0 48,71822 330 298 154 123 234 246.00 91 95 127.5 276.00 0 0 114 126 0 115.5 62.00 44,540 0 44,54023 330 298 154 123 234 246.00 91 0 117.00 276.00 0 0 114 126 0 52.00 28.00 39,355 0 39,35524 330 298 154 123 234 246.00 91 0 165.00 276.00 0 0 114 126 0 118.00 62.00 43,136 0 43,136

Total ($) 994,886 19,504 1,014,390

K.Chandrasekaran

etal./Inform

ationSciences

249(2013)

67–84

77



LR [10] 1,033,901AHN [10] 1,027,380LRAHN [10] 1,026,709EMO-ALHN [8] 1,016,062BRCFF 1,014,390

Table 9UCP status and hourly cost-26 unit system.

Hour Unit status 1, 2, . . . , 26 Total operating cost ($)

1 00000000000001001111000111 15522.492 00000000000001001111000111 15647.573 00000000000001001111000111 15068.534 00000000000000001111000111 14108.475 00000000000000001111000111 14108.476 00000000000000101111000111 16013.527 00000000011110101111000111 19509.228 00000000011111101111100111 25408.759 00000000011111111111110111 30127.510 10000000011111111111110111 31167.0311 00000000011111111111110111 29452.5912 10000000011111111111110111 30979.4813 11000000011110111111110111 29327.4314 11100000011110111111110111 28848.9815 11111000011110011111110111 27626.5516 00100000011110011111110111 26561.2717 00000000011110111111110111 27778.9218 00000000011110111111110111 27824.6319 00000000011111111111110111 29760.9820 00000000011111111111110111 30378.1221 00000000011111111111110111 29636.622 10000000011111111111100111 27027.9723 00000000011111111111000111 22296.9424 00000000000001111111000111 18756.76

Total cost ($) 582,938



ANN-DP [52] 613653.60ILR [30] 594116.50IPL-ALH [50] 583379.50BRCFF algorithm 582938.00


TOC obtained is $582938.00. Table 10 provides a comparison of the TOC obtained from the BRCFF algorithm with that ofother techniques available in the literature. The minimum cost produced by BRCFF is less than the results reported inliterature.


The 38 generating unit test system is adapted from [21]. The fuel cost coefficient and load data are taken from the samereference. The application is executed under the same conditions taken by reference [21] (i.e., the reserve requirement was11% of the hourly load, the start-up cost is constant and the ramp constraints are considered).

Out of 30 trials, the best TOC obtained for the 38 unit system for a 24-h time interval is $197082679.70. The detailedstatus and operating cost of generating units at each hour are given in Table 11. Table 12 provides a comparison of theTOC obtained from the BRCFF algorithm with that of other techniques. The minimum cost to date reported in the literatureis $197.41 [39], which is $327,320 higher than that obtained from BRCFF.

Table 11UCP status and hourly cost-38 unit system.

Hour Unit status 1, 2, . . . , 38 Total operating cost ($)

1 11111111010010000001111011000000000010 6452322.22 11111111010010000001111011000000000010 6091785.83 11111111010010000001111011000000000010 5,818,9914 11111111010010000001111011000000000010 5494396.95 11111111010010000001111011000000000010 5602777.56 11111111010010000001111011000000000010 5445769.17 11111111010010000001111011000000000010 5495104.48 11111111010010100001111011000000000010 6,775,9429 11111111111111110001111011000000000010 10,754,82410 11111111111111111001111011000000000010 10,036,95011 11111111111111111001111011000000000010 10,190,99612 11111111111111111001111011000000000010 10,337,71313 11111111111111111001111011000000000010 8667803.614 11111111111111111011111011000000000010 10,722,45015 11111111111111111001111011000000000010 10,569,15316 11111111111111111001111011000000000010 10,193,48717 11111111111111111001111011000000000010 9885858.518 11111111110111111001111011000000000010 8812379.719 11111111110111101001111011000000000010 8325098.520 11111111110111101001111011000000000010 9019449.321 11111111110110101001111011000000000010 8679161.122 11111111110110001001111011000000000010 8217181.723 11111111110110001001111011000000000010 7859428.224 11111111010110001001111011000000000010 7633658.3

Total cost ($) 197,082,680


Solution technique Minimum operating cost (Million$))

SA [14] 215.60DP [14] 215.20LR [14] 214.50FO [12] 213.90CLP [14] 213.80MRCGA [45] 206.70MACO [39] 203.32FAPSO [36] 199.07Twofold SA [37] 198.86ASSA [35] 198.84EMO-ALHN [9] 198.40HASSA [38] 197.41BRCFF algorithm 197.08


7.7. Solution quality and computation efficiency of FF algorithm

To validate the solution quality and computational efficiency of the proposed BRCFF algorithm, the UCP is solved for allthe test systems using standard GA and PSO algorithms. The comparison of computational efficiency is given in Table 13. Inall three methods (GA, PSO, and BRCFF), the maximum number of iterations is fixed as 300. It should also be noted that theinitial random generated population is considered the same for all three techniques, such as GA, PSO, and BRCFF.

Tables 4, 6, 8, 10, and 12 shows that the minimum TOC obtained by BRCFF is lower than the results reported in the exist-ing literature. Table 13 also shows that the proposed BRCFF algorithm has a better quality of solution with lesser computa-tional time than that of GA and PSO. The minimum and average TOC convergence characteristics of GA, PSO, and BRCFF for 3,12, 17, 26, and 38 unit test systems are shown in Figs. 4–8.

It is inferred from Figs. 4–8 that the characteristics of BRCFF algorithm steadily reach the minimum value and producebetter quality of solution for the test system with different sizes. However, in PSO and GA, the characteristics exhibit pre-mature convergence and settle to near-global optima compared to the BRCFF algorithm. Therefore, it can be concluded thatFF is computationally efficient and provides a quality solution with a computational time similar to GA and PSO.

Table 13Comparison of computation efficiencies.

Test system Solution technique Total operating cost $ Average computation time (sec)

Minimum value Average value

3 unit [13] GA 179221.86 179292.00 76PSO 178699.07 178749.00 91BRCFF 178611.57 178710.00 70

12 unit [13] GA 641786.59 643090.00 163PSO 641580.22 642400.00 186BRCFF 639938.60 641900.00 153

17 unit [10] GA 1021137.00 1023300.00 176PSO 1016594.00 1018500.00 191BRCFF 1014390.00 1016870.00 157

26 unit [44] GA 584569.00 600453.00 499PSO 583145.00 593060.00 516BRCFF 582938.00 588874.00 473

38 unit [21] GA 198076182.00 202691174.00 654PSO 197987586.00 201892590.00 678BRCFF 197082679.00 199071106.00 603

178400

178800

179200

179600

180000

0 50 100 150 200 250 300Iteration no.

Tota

l ope

ratin

g co

st($

) BRCFF-min BRCFF-avgPSO-min PSO-avgGA-min GA-avg

Fig. 4. Convergence graph of 3 unit system, min-minimum cost and avg-average cost.

638000

640000

642000

644000

646000

648000

0 50 100 150 200 250 300Iteration no.

Tota

l ope

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g co

st ($


Fig. 5. Convergence graph of 12 unit system.


In addition, the non-parametric test is carried out for the 38 unit system (largest system considered in the paper). Thistest is carried out to check the significance of the proposed FF algorithm for UCP with respect to GA and PSO. The systemdata is adapted from [21].

1008000

1018000

1028000

1038000

1048000

0 50 100 150 200 250 300Iteration no.

Tota

l ope

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g co

st ($

)

BRCFF BRCFFPSO PSOGA GA


580000585000

590000595000600000

605000610000615000

620000625000

0 50 100 150 200 250 300Iteration no

Tota

l ope

ratin

g co

st ($



194

198

202

206

210

214

218

0 50 100 150 200 250 300

Millions

Iteration no

Tota

l ope

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g co

st ($




To carry out non-parametric test, the UCP is solved for a 38 unit system for 30 trials. The minimum cost, average cost, andfrequency of achieving the minimum cost are better than the mean cost of FF algorithm, PSO, and GA as shown in Table 14.Fig. 9 shows the distribution of the TOC obtained from 30 trial runs.

In order to verify the significant improvements between the cost achieved by GA, PSO and BRCFF algorithms, Kruskal–Wallis one-way non-parametric analysis (one-way ANOVA statistical test) is carried out. The p-value obtained in the statisticaltest is about 1.82061 � 10�6 and 3.94251 � 10�5 for GA and PSO with respect to BRCFF algorithm, respectively. Since the p-values are very small, it can be concluded that the cost obtained by the proposed FF algorithm is significant compared to GAand PSO.

Table 14Performance comparison.

Parameters GA PSO BRCFF

Minimum cost ($) 198,076,182 197,987,586 197,082,679Average cost ($) 202,691,174 201,892,590 199,071,106Frequency of achieving minimum cost better than the mean cost 11 15 17

196000000

198000000

200000000

202000000

204000000

206000000

208000000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29Number of Trials

Tota

l ope

ratin

g co

st ($

)

Distribution of cost ($)-BRCFF Average cost ($)-BRCFFDistribution of cost ($)-GA Average cost ($)-GADistribution of cost ($)-PSO Average cost ($)-PSO

Fig. 9. Distribution of the cost of a 38 unit system.


8. Conclusion

In this paper, a new nature inspired algorithm, called the BRCFF algorithm, is efficiently and successfully implemented forsolving UCP by considering the system and generating unit constraints. Agents communicate with each other throughbioluminescence glowing which enables them to explore cost function space more effectively for solving UCP.

� The heuristic approach of the proposed binary-coded FF algorithm is efficient enough for scheduling generatingunits and the real coded FF algorithm is used for solving large-scale constrained EDP.

� A tanh function is introduced in the binary-coded FF algorithm to increase the probability of the flipping status ofthe binary variable, thereby improving the quality of solution and reducing the computational time of UCP.

� The proposed methodology is efficient enough to handle a large number of constraints in a large scale power systemoptimization problem.

� The proposed BRCFF algorithm obtains less cost than other methods available in the literature when validated onthe test systems such as 3, 12, 17, 26, and 38 unit systems. We also show that FF provides robust and accurate per-formance even for a difficult large-scale UCP.

The method is simple, easy to implement, and applicable for any large-scale system. From the results obtained, it is con-cluded that the BRCFF algorithm is a promising technique that exhibits certain superior characteristics compared to othertechniques currently available in the literature.

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binary real coded firefly algorithm for solving unit commitment problem

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