Multi-objective scheduling problem: Hybrid approach using fuzzy assisted cuckoo search algorithm

Swarm and Evolutionary Computation 5 (2012) 1–16

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Regular paper

Multi-objective scheduling problem: Hybrid approach using fuzzy assisted cuckoo search algorithm K. Chandrasekaran ∗ , Sishaj P. Simon Department of Electrical and Electronics Engineering, National Institute of Technology, Tiruchirappalli-620015, Tamil Nadu, India

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Article history: Received 4 August 2011 Received in revised form 21 October 2011 Accepted 12 January 2012 Available online 25 January 2012 Keywords: Binary and real coded cuckoo search algorithm Emission constraints Expected energy not supplied Fuzzy set theory Multi-objective unit commitment problem

abstract This article proposes a hybrid cuckoo search algorithm (CSA) integrated with fuzzy system for solving multi-objective unit commitment problem (MOUCP). The power system stresses the need for economic, non-polluting and reliable operation. Hence three conflicting functions such as fuel cost, emission and reliability level of the system are considered. CSA mimics the breeding behavior of cuckoos, where each individual searches the most suitable nest to lay an egg (compromise solution) in order to maximize the egg’s survival rate and achieve the best habitat society. Fuzzy set theory is used to create the fuzzy membership search domain where it consists of all possible compromise solutions. CSA searches the best compromise solution within the fuzzy search domain simultaneously tuning the fuzzy design boundary variables. Tuning of fuzzy design variables eliminate the requirement of expertise needed for setting these variables. On solving MOUCP, the proposed binary coded CSA finds the ON/OFF status of the generating units while the real coded CSA solves economic dispatch problem (EDP) and also tunes the fuzzy design boundary variables. The proposed methodology is tested and validated for both the single and multiobjective optimization problems. The effectiveness of the proposed technique is demonstrated on 6, 10, 26 and 40 unit test systems by comparing its performance with other methods reported in the literature. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The unit commitment problem (UCP) has an important role in daily operation of power system and determines the scheduling of generating units in a utility for minimizing the operating cost while satisfying the system and unit constraints. Due to the increased awareness of limiting environmental pollution caused by thermal power plants, many researchers treated emission as a constraint in a single objective UCP [1–3]. However the limitation of this approach in obtaining a trade-off solution, between the cost and emission which are conflicting in nature, cannot be minimized simultaneously. Hence the environmental/economic constrained UCP is a large-scale non-linear multi-objective optimization problem. In recent years, reliability of power system is a major focus of the system operators. Power system operators have to keep a certain amount of generation capacity as spinning reserve during scheduling operation. Thereby it ensures to withstand the sudden outage of some generating units/transmission lines or an unforeseen increase in load without having to resort to load

∗

Corresponding author. Tel.: +91 9095039194. E-mail addresses: [email protected] (K. Chandrasekaran), [email protected] (S.P. Simon). 2210-6502/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.swevo.2012.01.001

shedding. The traditional approach using deterministic criterion for setting the minimum amount of spinning reserve should be at least equal to the capacity of the largest unit, or to a specific percentage of the hourly system load. Even though the deterministic criterion is easy to implement, it does not match both the stochastic nature of the problem and the intrinsic reliability of each scheduled generator. Therefore it leads to inconsistent decision and variable operating risk levels. During the last 40 years, numerous techniques and methods have been developed to incorporate probabilistic reserve criteria in the formulation of the reserve constrained UCP [4–9]. Later methods provide promising result in the evaluation of spinning reserve by including various system risks. The choice of appropriate values for setting probabilistic reserve criterion depends on the global preference information of the system operator or decision-maker. Therefore, fixing the limits of reliability level is again conflicting in nature in order to obtain a trade-off solution between cost and reliability level. Hence maximizing the reliability level has to be handled as another objective function simultaneously in addition to cost and emission. Since many conflicting functions are to be handled, the system becomes highly non-smooth in nature which motivate the use of non-conventional cuckoo search algorithm. Many solution strategies are available to solve this highly nonsmooth UCP. A literary review of UCP and the solution techniques are given in [10,11]. Techniques like priority list method [12],

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Nomenclature Fc ai , bi , ci ei , fi E

Fuel cost function, ($) Cost coefficient of ith generator unit Valve point coefficient of ith generator unit Emission function, (ton) αi , βi , γi Emission coefficient of ith generator unit di , δi Exponential emission coefficient of ith generator unit H Total number of hours considered I i ,k Status of unit i at kth hour. (i.e.,) 1 for ON and 0 for OFF EENS k Reliability function (expected energy not supplied), (MW h) Loadk Total system demand at kth hour Li Load curtailed due to generator contingency i Pi,k Generation power output of unit i at kth hour pi Probability of unavailability for state i Pi,max Maximum power output of unit i Pi,min Minimum power output of unit i PLk System losses at kth hour RU(i) Ramp up rate limit of unit i RD(i) Ramp down rate limit of unit i T on (i) Minimum ON time for unit i T off (i) Minimum OFF time for unit i X on (i, k) Time duration for which unit i is ON at kth hour X off (i, k) Time duration for which unit i is OFF at kth hour x1 , x2 Fuzzy boundary values for each function Fc and E xmin , xmax Minimum and maximum value of fuzzy domain variables xa , xb , xc , xd Fuzzy boundary values for EENS Index i k

Generating unit index (subscript) Time index (subscript)

Abbreviation LC

Total number of contingencies leading to load curtailment.

dynamic programming [13], mixed integer programming [14], branch and bound [15,16] and Lagrangian Relaxation [17] are the widely used conventional techniques. The priority list method is simple and fast. However, it produces sub-optimal solution with higher operation cost. Dynamic programming method suffers from dimensionality problem. That is, with increase in problem size, the solution time increases rapidly with the number of generating units to be committed. Though LR method provides a fast solution, it suffers from numerical convergence and, the solution quality due to the dual nature of the algorithm is poor. In branch-and-bound method and mixed integer programming, the computational time increases enormously for a large-scale power system. So artificial intelligence techniques like, neural networks [18], expert systems [19], genetic algorithm (GA) [20–22], simulated annealing (SA) [8,23], evolutionary programming (EP) [24], tabu search [25], fuzzy logic [26], particle swarm optimization (PSO) [27–29], ant colony optimization (ACO) [30,31], frog leaping algorithm [32] and hybrid algorithm [33,34] are used. These are population based search techniques that search for the global or near global optimal solution, for any large-scale system incorporating all system constraints, with ease. In expert system, interaction with the plant operators are required making it inconvenient for a realistic system. The population based approach

like GA, EP, SA, PSO and ACO are able to obtain near optimal solution, for a large scale power system, but the computational time is quite high. Though many techniques are developed to solve UCP, no technique has been accepted as the best so far. In this context, an attempt is made to solve both the single and multi-objective EDP/UCP using a hybrid cuckoo search algorithm (CSA) integrated with fuzzy system. 2. Proposed work The aim of this paper is to show the efficiency of the binary and real coded cuckoo search algorithm [35,36] for solving EDP and UCP, respectively. Here, a tanh function is introduced in the binary coded CSA and an improvement in the performance is shown with respect to other methods available in the literature. The proposed formulation of MOUCP eliminates the role of the decision maker (system operator) in setting the spinning reserve constraints (either by deterministically or probabilistically) and emission constraints in UCP. Similar to other evolutionary methods, CSA starts with an initial fixed number of host nests (nests built by birds other than cuckoo) where each nest is assumed to contain a cuckoo’s egg. At the end of every generation of cuckoos, the size of host nests increases when compared to the initial number of nests. Then the fuzzy fitness is used to pick up the best compromise solution (which is equal to the initial number of host nests) that would be alive for the coming generations. At the end of every generation, the host bird will destroy the nest or abandon the solution (nest) which is a far away from the best solution. This is explained in Section 6 in detail. Fuzzy provides an excellent framework for integrating mathematical and heuristic approaches in order to obtain a more realistic problem formulation. Several attempts, employing fuzzyembedded features, are made in the past [37,38] for solving multi-objective problems. In [37,38], a fuzzification mechanism is introduced for the selection of global best position (gbest) from a population of candidate solutions. Here ‘gbest’ is not considered as one point solution but as an area. Therefore, each point in the area has different possibilities of being chosen as ‘gbest’. Here, the choice of appropriate setting of the boundary values in the fuzzy domain depend on the global preference information of the decision-maker and are similar to the weighted sum method [39,40]. Hence, in this paper, a new idea is proposed to tune the boundary values of the fuzzy domain using real coded CSA, thereby the fitness of nest is evaluated at every generation. In this article, the binary coded CSA is used to obtain the unit commitment scheduling for 24 h and the real coded CSA is used to tune the fuzzy design variables and to obtain the economic dispatch solution. The rest of the sections are organized as follows: In Section 3, the problem formulation of MOUCP is presented. Sections 4 and 5 describe the process of multiobjective optimization and the formulation of fuzzy membership function for different objectives respectively. Sections 6 and 7 explain the basic behavior of CSA and implementation of CSA for UCP. In Section 8, the effectiveness of the proposed approach is demonstrated on different test system and the results are discussed. Finally, the conclusion is given in Section 9. 3. Problem formulation Generally the unit commitment problem is solved for minimizing the total fuel cost. However in this paper, the UCP is formulated as a multi-objective optimization problem (MOOP) with three objectives such as fuel cost, emission of the generating units and reliability of the system. Modeling of fuel cost function, emis-

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sion function and reliability of the system for MOUCP is presented below: 3.1. Fuel cost function The fuel cost minimization problem is formulated as: Minimize H

Fc =

N

H



Fc (Pi,k ) =

k=1 i=1

3.4.6. Fuzzy design boundaries a. Fuzzy domain boundaries for Fc and E xmin < x1 < xmax

(11a)

x1 < x2 < xmax .

(11b)

b. Fuzzy domain boundaries for EENS

N

 ((ai + bi · Pi,k + ci · Pi,k 2 ) k=1 i=1

+ |ei · sin(fi · (Pi,min − Pi,k ))|).

3

(1)

xmin < xa < xmax

(12a)

xa < xb < xmax

(12b)

xb < xc < xmax

(12c)

xc < xd < xmax .

(12d)

3.2. Emission function Though classical dispatch is beneficial in terms of operating cost, fossil fuel based power plant tend to produce high SOx and NOx emissions. The emission from each unit depends on the power generated by that unit and can be modeled as a sum of a quadratic and an exponential function which is given in (2). E =

H  N 

E (Pi,k ) =

k=1 i=1

H  N  ((αi + βi · Pi,k + γi · Pi,k 2 ) k=1 i=1

+ δi · exp(di · Pi,k )).

(2)

3.3. Reliability function The reliability of the system plays a vital role in daily power system operation. Hence in UCP it is necessary to consider uncertainties in spinning reserve (SR) calculation. The system reliability level is dependent on the allocation of spinning reserve. Hence, to minimize EENS, enough spinning reserve has to be scheduled for each hour. Reliability level of the power system at each hour is calculated using (3) Min EENS =



p i Li ,

k ∈ [1, H ] (MW h)

(3)

i=LC

3.4. Problem constraints 3.4.1. Power balance constraint N  (Pi,k ∗ Ii,k ) = Loadk + PLk ,

k ∈ [1, H ].

(4)

i =1

3.4.2. Spinning reserve constraint N  (Pi,max ∗ Ii,k ) ≥ Loadk + PLk ,

k ∈ [1, H ].

(5)

i =1

(6)

3.4.4. Unit minimum ON/OFF durations

[X on (i, k) − T on (i)] ∗ [Ii,k−1 − Ii,k ] ≥ 0

(7)

[X (i, k − 1) − T (i)] ∗ [Ii,k−1 − Ii,k ] ≥ 0.

(8)

off

off

Many real world optimization problems involve simultaneous optimization of several conflicting objectives. Multi-objective optimization problems with such conflicting objectives give rise to a set of optimal solution, rather than a single optimal solution, since no solution can be considered to be better than other solutions without adequate information. These set of optimal solutions are called as Pareto-optimal solutions [41]. A general multiobjective optimization problem consists of multiple objectives to be optimized simultaneously with various equality and inequality constraints. This can be generally formulated as Min fi (y),

i = 1, 2, . . . , NO gj (y) = 0, hk (y) ≤ 0,

 subject to :

(13)

j = 1, 2, . . . , M k = 1, 2, . . . , K

(14)

where fi is the ith objective function, y is a decision vector that represents a solution, NO is the number of objective functions, M and K are the number of equality and inequality constraints respectively. Generally, the evaluation functions are contradicting with each other. Thus Pareto-dominance is used to determine an optimal solution. For a multi-objective optimization problem, any two solutions y1 and y2 can have any one of two possibilities, one dominates the other or none dominates any other possibility. In a minimization problem, without loss of generality, solution y1 dominates y2 if the following conditions are satisfied. 1. ∀i ∈ {1, 2, . . . , NO} :

fi (y1) ≤ fi (y2)

(15)

2. ∃j ∈ {1, 2, . . . , NO} :

fj (y1) < fj (y2).

(16)

If any one of the above conditions is violated, then the solution y1 does not dominate y2. If y1 dominates solution y2, y1 is called as the non-dominated solution. The solutions that are non-dominated within the entire search space are denoted as Pareto-optimal solutions. 5. Fuzzy membership function formulation

3.4.3. Capacity limit Pi,min ≤ Pi,k ≤ Pi,max .

4. Formulation of a multi-objective function

3.4.5. Unit ramp constraints Pi,k − Pi,k−1 ≤ RU(i)

(9)

Pi,k−1 − Pi,k ≤ RD(i).

(10)

Pareto optimal concept using fuzzy membership is used to evaluate the fitness of each nest. In the fuzzy domain, it is required to formulate a fuzzy membership function for each objective, i.e., the corresponding membership function value should indicate the associated degree of satisfaction for that objective. The formulation is done using quasi-optimization solutions obtained by evaluating each objective function individually as a single objective optimization problem. The single-objective optimization of the Fc , E and EENS, known as quasi-optimization, can be expressed as in Eq. (17). J1 = min ·Fc ;

J2 = min ·E ;

J3 = min ·EENS .

Subject to: the operating constraints given in Eqs. (4)–(10).

(17)

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Fig. 1. Fuzzy membership functions for Fc and E.

Fig. 2. Fuzzy membership function for EENS.

5.1. Fuzzy membership functions for Fc and E In the proposed methodology fuzzy works as an assisted function to Binary and Real coded CSA (BRCSA). Therefore fuzzy membership function should be framed in such a manner that it aids BRCSA in maximizing the fitness function. The membership function chosen for objective function Fc and E is same and is shown in Fig. 1. Here the x1 and x2 are the boundary values in the fuzzy domain. The design of the membership function implies that for any solution, if the objective function (Fc and E) in the fuzzy domain is more than x2 , then the associated fuzzy membership function value is assigned to be zero. Such solutions (corresponding nest positions) do not maximize the fitness function when evaluated using the BRCSA and do not participate in the optimal solution set. On the other hand, if the objective function is less than x1 , then the associated fuzzy membership function value is assigned to be unity. Such solutions do participate and maximize the fitness function. If the objective function in the fuzzy domain is between x1 and x2 , then the associated fuzzy membership function value is computed using Eq. (18), and such solutions will participate in the optimization process depending on the fitness value.

 1,    (xj − x1 ) p µj = ,  (   x2 − x1 ) 0,

for xj < x1 ; for x1 ≤ xj ≤ x2 ;

(18)

for xj > x2

where j ∈ {Fc , E }, xj is the degree of the objective function in the p fuzzy domain and µj is the membership function value for the objective function. 5.2. Fuzzy membership function for EENS The membership function chosen for the EENS function is shown in Fig. 2. Similar to Section 5.1, minimum and maximum value of EENS function is calculated using the quasioptimization solution in fuzzy domain. Here the xa , xb , xc and xd are the boundary values in the fuzzy domain. The design of the membership function implies that, for an objective function of EENS in the fuzzy domain which is less than xa , the system cost is increased proportionally. Hence the associated fuzzy membership function value is assigned to be zero. Such solutions (corresponding nest positions) do not maximize the fitness function when evaluated using the BRCSA and do not participate in the optimal solution set. On the other hand, when the objective function of EENS in fuzzy domain is greater than xd, it will affect the system reliability and hence will not take part in the optimal solution set. Therefore xa and xd value should be chosen based on the system generation capacity. If the value of the objective function in the fuzzy domain is between xb and xc , then the associated fuzzy membership function is set to be unity and such solutions do participate in maximizing the fitness function. If the membership value is within

[xa , xb ] or [xc , xd ], then the associated fuzzy membership function value is computed using Eq. (19) and such solutions will participate in the optimization process depending on the fitness value. 0,    (xj − xa )    ,   (xb − xa ) p µr = 1,    (xj − xc )   ,    (xd − xc ) 0,

for xj < xa ; for xa ≤ xj ≤ xb ; for xb < xj < xc ;

(19)

for xc ≤ xj ≤ xd ; for xj > xd ;

p

where µr is the membership function value for the objective function EENS. 6. Overview of cuckoo search algorithm In this section, the breeding behavior of cuckoos and the characteristics of Levy flights in reaching best habitat societies are discussed. Cuckoo search algorithm (CSA) is one of the most recently defined algorithms [35,36] by Xin-She Yang and Suash Deb in 2009. It has been developed by simulating the intelligent breeding behavior of cuckoos. It is a population-based search procedure that is used as an optimization tool, in solving complex, nonlinear and non-convex optimization problems. Female parasitic cuckoos specialize and lay eggs that closely resemble the eggs of their chosen host nest (i.e., nest is built by other species bird). Cuckoo chooses this host nest by natural selection. The shell of the cuckoo egg is usually thick. They have two distinct layers with an outer chalky layer that is believed to provide resistance to cracking when the eggs are dropped in the host nest. The cuckoo egg hatches earlier than the host bird’s egg, and the cuckoo chick grows faster. Alien eggs (i.e., remaining cuckoo’s egg in the nest) are detected by host birds with a probability Pa ∈ [0, 1]and these eggs are thrown away or the nest is abandoned, and a completely new nest is built, in a new location by host bird. The mature cuckoo form societies and each society have its habitat region to live in. The best habitat from all of the societies will be the destination for the cuckoos in other societies. Then they immigrate toward this best habitat. A randomly distributed initial population of host nest is generated and then the population of solutions is subjected to repeated cycles of the search process of the cuckoo birds to lay an egg. The cuckoo randomly chooses the host nest position to lay an egg using Levy flights random-walk and is given in Eqs. (20)–(21). Vpq t +1 = Vpq t + spq × Levy(λ) × α

     Γ (1 + λ) × sin π ×λ 2   Levy(λ) =   (λ−1)  Γ (1+λ) ×λ×2 2 2

(20)

1 λ    

(21)

where λ is a constant (1 ≤ λ ≤ 3), t is the current generation number and α is a random number generated between [−1, 1].

K. Chandrasekaran, S.P. Simon / Swarm and Evolutionary Computation 5 (2012) 1–16

spq = Vpq t − Vfq t

(22)

where p, f ∈ {1, 2, . . . , m} and q ∈ {1, 2 . . . D} are randomly chosen indexes. Although f is determined randomly, it has to be different from p. D is the number of parameters to be optimized and m is the total population of host nest positions. Using (20) the cuckoo chooses the host nest or communal nest and an egg laid by a cuckoo is evaluated. The host bird identifies the alien egg with the probability value associated with that quality of an egg using Eq. (23). Prop = (0.9 ∗ Fit p / max(Fit )) + 0.1

xp = xp min + rand (0, 1) × (xp max − xp min )

(24)

where xp min and xp max are the minimum and maximum limits of the parameter to be optimized. 7. Implementation of binary coded CSA for UCP In UCP, binary numbers 1 and 0 are used to indicate the unit status ON/OFF. The CSA used in Refs. [35,36] is essentially a realcoded algorithm, thus some modifications are needed to deal with binary-coded optimization problem. Cuckoo randomly chooses the host nest position, to lay an egg, using Levy flights random-walk using Eq. (20). In (20), the α is a random number generated in the range [−1, 1] and λ is between the range of 1 and 3. When λ is varied between 1 and 3, the corresponding variation in (21) is between 0 and 1. From these values, it is observed that the Levy flights random-walk given in (20) is varied between −1 and 2. Generally to accomplish this, sigmoid function is used [26] as in Eq. (25). 1 1 + exp(−Vpq )

.

(25)

However, to improve the performance of binary CSA, another function called tanh is proposed and is given in (26). f (Vpq ) = tanh (|Vpq |) =

exp(2 ∗ |Vpq |) − 1 exp(2 ∗ |Vpq |) + 1

0.8 0.6 0.4

tanh function sigmoid function

0.2 0 -1

-0.5

0

0.5

1

1.5

2

Vpq

Fig. 3. Variation of Vpq and the function of Vpq values.

(23)

where Fit p is the fitness value of the solution p which is proportional to the quality of an egg in the nest position p and Prop gives the survival probability rate of the cuckoo’s egg. If the random generated probability Pa ∈ [0, 1] is greater than the Prop , then the alien egg is identified by the host bird. Then the host bird, destroys the alien egg away or abandon the nest and, cuckoo find a new host’s nest (in new position) using Eq. (24) for laying an egg. Otherwise, the egg grows up and be alive for the next generation based on the fitness function.

f (Vpq ) =

1

f(Vpq)

Also s > 0 is the step size which should be related to the scales of the problem of interests. If s is too large, then the new solution generated will be too far away from the old solution (or even jump out of the bounds). Then, such a move is unlikely to be accepted. If it is too small, the change is too small to be significant, and consequently such search is not efficient. So a proper step size is important to maintain the search as efficiently as possible. Hence the step size is calculated using Eq. (22).

5

.

(26)

Both the functions, scales the Vpq value within [0, 1] as shown in Fig. 3. A random number is generated between 0 and 1 to decide the unit status as 0 or 1. If f (Vpq ) is greater than rand (0, 1), then the unit status is 1 otherwise 0. The chances of bit flipping are found to be more in case of tanh function than in sigmoid function. Based on 20 trial runs, it is observed that the performance of tanh function on reaching quality solution is faster when compared to sigmoid function (given in Section 8.2.1).

Fig. 4. Initialization of binary string population.

7.1. Initial generation of binary string population (binary coded CSA) Randomly generate a population of M initial solutions represented by a binary string. Initialize randomly an initial population M = [X1 ; X2 ; X3 ; . . . ; Xm ] of m solutions or host nest positions in the multi-dimensional solution space where m represents the population of host nest. Each solution of X is represented by the D-dimensional vector. Here D is equal to N ∗ H. A population of M initial solution with D dimensional vector is shown in Fig. 4. 7.2. Repair strategy for constraint management in binary coded CSA Whenever the commitment status for each time interval is generated randomly or by the modification of nest position, violation of minimum up/down time constraints (7)–(8) and spinning reserve constraint (5) has 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)–(8) are checked for each unit. If there is any violation in the minimum up or down time constraint then a repair scheme is performed to overcome the violation. For instance, let us assume that the T on and T off for a hypothetical unit is 4 and 5. For a scheduling interval of 12 h, if the actual off time for that unit is 3 h (5th–7th hour), then it violates the T off constraint. In this case, the unit status before 5th hour or after 7th hour should be made OFF. By doing this change, if it violates the T on constraint, then the status of the units are made ON corresponding to the unit status. Step 4: The repair scheme in step 3 may affect the spinning reserve constraint of the system. If the reliability level is met, then accept the feasible solution. Otherwise go to step 1. A minimum number of trials (approximately 10 times) should be set for the repair mechanism. These steps are carried out for the entire hourly load. This is represented in Fig. 5.

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Random generation of initial solution as binary string

Is constraints (5) satisfied

No

Commit additional units

Yes

Yes

Is constraints (7-8) satisfied

No

Do repair strategy

Feasible solution

Step 5: If there is any violation in constraint (12), a repair scheme given in step 5a is performed to overcome the violation otherwise go to next step. Step 5a: If xa < xmin , then do xa = xmin + (xmax − xmin ) × xa . Otherwise go to step 5b. Step 5b: If xb < xa , then do xb = xa +(xmax − xa )× xb . Otherwise go to step 5c. Step 5c: If xc < xb , then do xc = xb +(xmax − xb )× xc . Otherwise go to step 5d. Step 5d: If xd < xc , then do xd = xc +(xmax − xc )× xd . Otherwise go to step 6. Step 6: End. 7.5. Fuzzy fitness evaluation

Fig. 5. Flowchart on the repair strategy.

A fuzzification mechanism and fitness sharing is introduced in CSA to pick out the best compromise solution (nest) that will be alive for the next generation. For each non-dominated solution, the normalized membership fuzzy fitness function (Fit) is calculated using Eq. (27) Fit p =

(µc p + µe p + µr p )  (µc p + µe p + µr p )

(27)

p=1 to m

Fig. 6. Initialization of real string population.

7.3. Initialization of real string population (real coded CSA) Randomly generate a population of R initial solutions represented by real values for unit status of the pth population with S dimensional vector. This representation is shown in Fig. 6. It should be noted that the real values will be initialized only for the unit which has ON status obtained by binary coded CSA. Hence initialize randomly an initial population R = [Y1 ; Y2 ; Y3 ; . . . ; Ym ] of m solutions or host nest positions in the multi-dimensional solution space where m represents the population of host nest. Each solution of Y is represented by the S-dimensional vector. Here S is equal to (Non ∗ H ) + NF . Non is the total number of ON generating units. NF is the number of boundary values in the fuzzy domain and Bi is the fuzzy domain boundary values. 7.4. Repair strategy for constraint management in real coded CSA Whenever there is a modification of nest position by real coded CSA, violation of capacity limits of generating units (6) and fuzzy design boundary constraints (11)–(12) has to be checked as follows: Step 1: If the generated power (Pi ) capacity limits (6) is met, then go to step 3. Otherwise, go to next step. Step 2: If Pi > Pmax then Pi = Pmax . If Pi < Pmin then Pi = Pmin . Then go to step 3. Step 3: If the limits of the design variables (11)–(12) are met, then go to step 6. Otherwise, go to next step. Step 4: If there is any violation in constraint (11), a repair scheme given in step 4a is performed to overcome the violation, otherwise go to next step. Step 4a: If x1 < xmin , then do x1 = xmin + (xmax − xmin ) × x1 and go to next step. Step 4b: If x2 < x1 , then do x2 = x1 + (xmax − x1 ) × x2 and go to step 5.

where m is the total number of non-dominated solutions or population of the host nests. The best compromise solution is the one that has the maximum value in the population of host nests i.e., the nest having highest quality of egg compared with other nests. The step-by-step procedure for the proposed method is given as a flowchart in Fig. 7. Graphical representation of flow of algorithm is given in Fig. 8. For easy understanding, a sample problem is taken and the computational flow of algorithm is explained in Appendix. 8. Results and discussions All the programs are developed using MATLAB 7.01. The system configuration is Pentium IV processor with 3.2 GHz speed and 1 GB RAM. Different test cases are carried out to show the efficiency of the proposed CSA and hybrid methodology which are given below. Case 1: To validate the real coded CSA, EDP is solved for case 1 and the results are compared with Refs. [42–52]. Case 2: To validate binary coded CSA and the performance of tanh function, UCP is solved for two test systems and the results are compared with Refs. [53–62] Case 3: To validate hybrid approach of CSA and tuned fuzzy system, two objective functions (i.e., fuel cost and emission function) for multi-objective economic dispatch problem (MOEDP) is solved and the results are compared. Case 4: Finally the proposed methodology is applied for three objective functions (i.e., fuel cost, emission and reliability level) for solving MOUCP.

8.1. Case 1: 40 unit system In this case, to validate the applicability of the real coded CSA, EDP is solved for 40 unit system. Cost curves include valvepoint effects and the test data are adapted from [42] and the load of the system is taken as 10 500 MW. Out of 20 trials, the best total production cost using the proposed method is obtained as $121 414.6703. Table 1 provides the comparison of total cost obtained from real coded CSA with respect to other available techniques. The corresponding dispatch of units is shown in Table 2. It is clearly seen that the proposed method yields better results than other techniques so far reported in the literature.

K. Chandrasekaran, S.P. Simon / Swarm and Evolutionary Computation 5 (2012) 1–16

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SolvingEDP and Tuning fuzzy design variablesusing Real codedCSA

Start

Start

Generate random initial real coded population as given in section 7.3 for the status ofpth population

Input system data, set binary coded CSA parameters Generate random initial population M as given in section 7.1

Setgen=1 Total number of host nest ism Modification of host nest to lay an egg by Levy flights random-walk using (20)

Set gen=1 Total number of host nest ism Modification of host nest to lay an egg by Levy flights random-walk using (20)

Do repair strategy for constraint management as given in section 7.4 Calculate objectives and Evaluate fitnessFit of the modified position using (27) and calculatePro using equation (23)

Do repair strategy for constraint management as given in section 7.2 Get fitnessFitof the modified position with unit status and fuzzy boundaries. Then calculatePro using equation (23)

?

Yes

Find a new host nest using equation (24) for laying an egg Add the new nest with the previous population of host nests Total number of host nest is > =m& <=2m Do repair strategy for constraint management as given in section 7.2

?

Yes

For each nest position If host bird discovered the alien egg with probabilityPa i.e. ifPap>Prop

Total number of nest is m

No

No

Find a new host nest using equation (24) for laying an egg

Calling Real coded CSA

Total number of host nest is m

Egg grows up in the same nest for next generation

For each nest position If host birddiscoveredthe alien egg with probabilityPa i.e. ifPap>Prop

Egg grows up in the same nest for next generation

Add the new nest to the previous population of host nests Total number of host nest is >=m & <=2m Do repair strategy for constraint management as given in section 7.4 Calculate objectives. Then evaluate fitnessFit of the modified position using (27) SortFitin descending order, pick out firstm nest position to grow for next generation and abandon the remaining nests

Get fitness Fitof the modified position with unit status and fuzzy boundaries

Total number of nest ism Sort Fit in descending order, pick out firstm nest position to grow for next generation and abandon the remaining nests Total number of host nest ism Memorize the best solution at each interval gen=gen+1 Yes

Is gen<=genmax

Memorize the best solution at each interval

No Store the best solution with corresponding fitness value and fuzzy boundaries. Then dop=p+1

Is gen<=genrmax

Yes gen=gen+1

Yes Is p<=popmax

No End

Output the best solution

No Output the best solution of generatedUC status with fitness value and fuzzy boundaries

Fig. 7. Flowchart for MOUCP using hybrid methodology.

8.1.1. Effect of variation of CSA parameters Good converge behavior can be obtained if the three control parameters namely initial population of host nests, maximum generation limit and λ can be optimally tuned. Setting of these cuckoo parameters optimally would also yield better solution and lesser computational time. In order to avoid misleading results due to the breeding behavior of cuckoos in the CSA, several test runs are carried out to set the cuckoo parameters. 20 trial runs are carried out for the test case 1. The nest population size is then varied between 25 and 300 for the maximum generation of 500 and λ is set as 1. Fig. 9 shows the average value of the total operating cost out of 20 trials for different population of host nests. When

the nest population is greater than 100, there is no significant improvement in the average operating cost. Also, in general when there is an increase in the nest population size, the number of generation required by the CSA to converge to the optimum solution decreases. On the other hand, the CPU time required for the evaluation of generation increases almost linearly with the nest population which is shown in Fig. 9. Therefore an intermediate value of 100 is chosen for the initial population of host nests, which gives an increase in efficiency and helps in finding near global optimum solution. Thus the best possible setting for the population size is chosen as 100. When the nest population size and λ is set as 100 and 1 respectively, the maximum generation limit is varied between 50

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Fig. 8. Graphical representation of algorithm for MOUCP using hybrid methodology. Table 1 Comparison of results – 40 unit system – case 1. Solution technique

HGPSO [43] SPSO [43] PSO [44] CEP [44] HGAPSO [43] FEP [44] MFEP [44] IFEP [44] TM [44] MSL [44] EP–SQP [44] MPSO [44] ESO [44] HPSOM [43] PSO–SQP [44] PSO–LRS [45] Improved GA [46] HPSOWM [43] IGAMU [47] HDE [48] AIS binary [49] DEC(2)-SQP(1) [44] PSO [45] APSO(1) [45] ST-HDE [48] NPSO–LRS [44] APSO(2) [45] Continuous GA [49] SOHPSO [50] CSO [49] AIS real [49] ABC [49] BF [51] MBFA [52] Real coded CSA

Operating cost ($)

Running time (s)

Minimum

Mean

Maximum

Mean time

Best time

124 797.13 124 350.40 123 930.45 123 488.29 122 780.00 122 679.71 122 647.57 122 624.35 122 477.78 122 406.10 122 323.97 122 252.26 122 122.16 122 112.40 122 094.67 122 035.79 121 915.93 121 915.30 121 819.25 121 813.26 121 760.58 121 741.97 121 735.47 121 704.73 121 698.51 121 664.43 121 663.52 121 523.10 121 501.14 121 461.67 121 458.18 121 441.03 121 423.63 121 415.653 121 414.6703

126 855.70 126 074.40 124 154.49 124 793.48 124 575.70 124 119.37 123 489.74 123 382.00 123 078.21 NA 122 379.63 NA 122 558.45 124 350.87 122 245.25 122 558.45 122 811.41 122 844.40 NA 122 705.66 121 596.70 122 295.12 122 513.91 122 221.36 122 304.30 122 209.31 122 153.67 121 998.70 121 853.57 121 936.19 121 484.23 121 995.82 121 814.94 NA 121 496.21

NA NA NA 126 902.89 NA 127 245.59 124 356.47 125 740.63 124 693.81 NA NA NA 123 143.07 NA NA 123 461.67 123 334.00 NA NA NA 121 865.44 122 839.29 123 467.4 122 995.09 NA 122 981.59 122 912.39 122 753.87 122 446.30 122 844.53 121 502.83 122 123.77 NA NA 121 532.13

NA NA 933.39 1956.93 NA NA 2196.10 1167.35 94.28 NA 997.73 NA NA NA NA NA NA NA NA NA 844.11 14.26 NA NA NA 3.93 NA NA NA NA 268.37 32.45 NA NA 29

48.47 23.92 NA NA 24.22 1037.90 2194.70 1165.70 91.16 0.0078 NA NA 0.261 23.91 NA NA 15.75 25.39 27.03 6.92 839.52 NA NA NA 6.07 NA NA 1165.70 NA NA 264.40 30.02 NA NA 24

and 800. It is observed from Fig. 10 that, an increase in maximum generation limit leads to an improvement in the average cost. When the maximum generation is greater than 750, there is no significant improvement in the total operating cost. Thus the best possible setting for the maximum number of generation is set as 750 for the nest population size of 100. Finally, λ is varied from 0 to 1, when the nest population size and maximum generation limit is set as 100 and 750 respectively. Fig. 11 gives the minimum value obtained in 20 trials for different

values of λ. From Fig. 11, it is concluded that, the best λ value chosen is 1 for 40 unit system. Once the parameters are finalized for this system, the same settings can be used even if there is a change in the forecast load. 8.2. Case 2 In case 2, two test systems are taken to show the robustness and the performance of the tanh function when used in the BRCSA.

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Table 2 Optimal dispatch of generators – 40 unit system – case 1. Unit no.

Generated power (MW)

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40

110.8001 110.7998 97.3999 179.7327 87.8009 140.0000 259.6035 284.5993 284.5992 130.0000 168.7991 168.7999 214.7598 394.2798 394.2794 304.5198 489.2793 489.2791 511.2792 511.2796 523.2794 523.2807 523.2794 523.2794 523.2798 523.2794 10.0000 10.0000 10.0000 92.7510 190.0000 190.0000 190.0000 164.8003 164.7994 164.8004 110.0000 110.0000 110.0000 511.2792

Fig. 10. Variation of average operating cost with maximum generation limit— case 1.

Fig. 11. Variation of operating cost with lambda (λ)—case 1.

Fig. 12. Convergence graph for 10 unit system—case 2; (min—minimum cost, avg—average cost).

Fig. 9. Average operating cost for different initial populations of host nest—case 1.

8.2.1. Test system 1: 10 unit system To validate the performance of tanh function in binary coded CSA, a 10 unit system is solved for UCP. Here, EDP is solved using real coded CSA. The generator cost coefficient, ramp rate limit and power generation limits are adapted from Ref. [53]. The load profile for a period of 24 h is taken from the same reference and the system spinning reserve is considered as 10% of load at each hour. Here the efficiency of the binary CSA is compared with the binary PSO algorithm as in the Refs. [53–59]. In [53–59], the binary PSO solution is obtained using the sigmoid function which is given in Eq. (25). However in the proposed method, to improve the performance of the binary CSA, tanh function is used and the results are compared with Refs. [53–59].

Similar to case 1, the parameter tuning for binary coded CSA is carried out for 10 unit system and the final best combination of parameters are λ = 1, initial population of host nest = 100 and maximum generation limit = 300 for binary coded CSA. Similarly for real coded CSA, the best combination of parameter settings are λ = 1.5, initial population of host nest = 100 and maximum generation limit = 500. The total production cost obtained for the 10 unit system for a 24-h-time interval is $563 937. The detail of the power dispatch of each unit is given in Table 3. Table 4 provides the comparison of the total cost obtained from binary CSA using tanh with that of PSO using sigmoid function. It is clearly seen that the proposed tanh function in binary coded CSA yields better results than other techniques available in the literature. The minimum cost so far reported in literature is $563 942 [48] which is $5 higher than the optimum solution obtained from BRCSA. Though the amount saved is less significant, the BRCSA produces the least of the minimum production costs, average production costs and worst costs (Table 4) with respect to other methods available in the literature. It is inferred from Fig. 12, that the characteristics of binary CSA using tanh function steadily reaches the minimum value after few

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Table 3 UC status and power dispatch for 10 unit system—case 2. Hour

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

Total load (MW)

Fuel cost ($)

Startup cost ($)

Operating cost ($)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455

245 295 370 455 390 360 410 455 455 455 455 455 455 455 455 310 260 360 455 455 455 455 425 345

0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0

0 0 25 40 25 25 25 30 85 162 162 162 162 85 30 25 25 25 30 162 85 145 0 0

0 0 0 0 0 0 0 0 20 33 73 80 33 20 0 0 0 0 0 33 20 20 20 0

0 0 0 0 0 0 0 0 25 25 25 25 25 25 0 0 0 0 0 25 25 25 0 0

0 0 0 0 0 0 0 0 0 10 10 43 10 0 0 0 0 0 0 10 0 0 0 0

0 0 0 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0

700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500 1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

13 683 14 554 16 809 18 598 20 020 22 387 23 262 24 150 27 251 30 058 31 916 33 890 30 058 27 251 24 150 21 514 20 642 22 387 24 150 30 058 27 251 22 736 17 645 15 427

0 0 900 0 560 1100 0 0 860 60 60 60 0 0 0 0 0 0 0 490 0 0 0 0

13 683 14 554 17 709 18 598 20 580 23 487 23 262 24 150 28 111 30 118 31 976 33 950 30 058 27 251 24 150 21 514 20 642 22 387 24 150 30 548 27 251 22 736 17 645 15 427

559 847

4090

563 937

Total ($)

Table 4 Comparison of results—case 2. Solution technique

Minimum operating cost ($)

Mean operating cost ($)

Maximum operating cost ($)

PSO [53] MIPSO [54] APSO [55] MHPSO [56] IBPSO [57] QBPSO [58] HPSO [59] BRCSA

574 285.1 572 075.0 565 450.0 564 419.0 563 977.0 563 977.0 563 942.3 563 937.0

– – – 564 423.9 564 155.0 – 564 772.3 563 945.0

– – – – 565 312 – 565 785.3 565 503.0

iterations when compared to sigmoid function. It can be concluded that convergence characteristic of binary CSA using tanh function is fast than the binary CSA that uses sigmoid function. Hence the binary CSA using tanh function produces quality solution in less time when compared with other techniques. 8.2.2. Test system 2: 26 unit system A test system comprising of 26 generating units [60] is considered. The fuel cost coefficient and load profile is adapted from the same reference. Here two strategies of reserve scheduling are carried out to validate the proposed algorithm. Strategy 1

Strategy 2

The system spinning reserve is set to the maximum capacity of the largest committed unit (deterministic criterion) as in the Ref. [60] and results are compared with Refs. [60–62]. The system spinning reserve is set by the reliability index (probabilistic criterion) loss of load probability (LOLP) as in the Ref. [8].

In [8], LOLP index is used to evaluate the reliability of the system. Also loss of load probability (LOLP) index is expressed in % and the lead time of the system is fixed for 4 h. Hence in strategy 2, for the complete time horizon, the calculated reliability index (LOLP) value should be less than or less than equal to the specified LOLP value (LOLPspec ). Here LOLPspec is defined as 1.5%. UCP for both strategies of reserve scheduling are carried out on 26 unit system. The parameter tuning is carried out for 26 unit

system, and the final combination of parameters that provided the best results are λ = 1, initial population of host nest = 100 and maximum generation limit = 300. Similarly for the real coded CSA, the final combination of parameters that provided the best results are λ = 1.5, initial population of host nest = 100 and maximum generation limit = 500. Out of 20 trials, the best total production cost obtained for strategy 1 and strategy 2 is given in Table 5. The generating unit status and hourly operating cost for strategy 1 and strategy 2 is given in Table 5. Table 6 provides the comparison of the total operating cost obtained from BRCSA for strategies 1 and 2 with respect to other techniques existing in the literature. In both strategies, the minimum cost produced by BRCSA which is less than the results reported in the literature. 8.2.2.1. Solution quality and computational efficiency of CSA. To validate the computational efficiency of the proposed CSA, the UCP is solved for the 26 unit system using simple real coded GA [63] and PSO algorithm [64]. In all the three methods (GA, PSO and BRCSA), the maximum number of iteration is fixed as 300. Also it is to be noted that the initial random generated population is taken same for all the three techniques (GA, PSO and CSA). Tables 6 and 7 shows that the minimum cost produced by BRCSA which is less than minimum solutions reported in the existing literature. From Table 7, it can be seen that the time taken by the CSA is lesser than that of GA and PSO. Although the real coded GA has better computation time than PSO the phenomenon of premature convergence degrades its performance. The convergence characteristic of GA, PSO and BRCSA for the test system discussed here is shown in Figs. 13–14. It is inferred from Figs. 13–14, that the characteristic of CSA steadily reaches the minimum value after few iterations and produce better quality solutions. It can be concluded that CSA is computationally efficient providing quality solutions obtained in minimum computational time at par with GA and PSO. However, GA the characteristics exhibit premature convergence and settles to near-global optima compared to CSA. 8.3. Case 3: 6 unit system To compare the benefits of the proposed hybrid methodology, MOEDP (i.e., fuel cost and emission function) is solved using real

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Table 5 UCP status and hourly cost – 26 unit system – strategy 1 and strategy 2. Strategy 2—LOLP = 1.5%

Hour Strategy 1 Unit no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Fuel cost ($)

11111111111100000000000000 19 015 11111111111100000000000000 19 371 11111111110100000000000000 18 789 11111111110100000100000000 18 966 11111111111100000000000000 19 610 11111111111110000000000000 21 187 11111111111111000111000000 23 698 11111111111111111000000000 32 643 11111111111111111000000000 34 726 11111111111111111111000000 36 050 11111111111111111111101110 38 251 11111111111111111111000000 35 816 11111111111111111111000000 35 816 11111111111111111000000000 34 917 11111111111111111111110000 36 582 11111111111111111111101100 37 605 11111111111111111000000000 34 917 11111111111111111000000000 34 534 11111111111111111000000000 33 963 11111111111111111000000000 34 917 11111111111111111111000000 36 050 11111111111111111000000000 33 584 11111111111111100111000000 27 422 11111111111110000000000000 21 066 Total ($) 719 495

Startup cost ($)

Operating cost ($)

Unit no.

Fuel cost ($)

106.93 0 0 0 74.329 130.53 132.62 1048.3 0 0 119.75 0 0 0 0 74.587 0 0 0 0 0 0 0 0 1687.046

19 121.93 19 371 18 789 18 966 19 684.33 21 317.53 23 830.62 33 691.3 34 726 36 050 38 370.75 35 816 35 816 34 917 36 582 37 679.59 34 917 34 534 33 963 34 917 36 050 33 584 27 422 21 066 721 182

11111111100000000000000000 18 429 11111111100000000000000000 18 789 11111111000000000000000000 18 205 11111111010000000000000000 18 429 11111111010000000000000000 19 030 11111111111000000000000000 20 478 11111111111100000000000000 22 893 11111111111110110000000000 31 965 11111111111111110000000000 34 298 11111111111111111000000000 35 960 11111111111111111000000000 37 608 11111111111111111000000000 35 726 11111111111111111000000000 35 726 11111111111111110000000000 34 533 11111111111111110111110000 36 334 11111111111111110111111000 37 219 11111111111111110000000000 34 533 11111111111111110000000000 34 063 11111111111111110000000000 33 410 11111111111111110000000000 34 533 11111111111111110111000000 35 800 11111111111111110000000000 33 025 11111111111111000000000000 26 617 11111111110000000000000000 20 262 Total ($) 707 865

Startup cost ($)

Operating cost ($)

0 0 0 81.606 0 172.16 132.62 833.12 135.53 360.62 0 0 0 0 0 39.993 0 0 0 0 0 0 0 0 1755.649

18 429 18 789 18 205 18 510.61 19 030 20 650.16 23 025.62 32 798.12 34 433.53 36 320.62 37 608 35 726 35 726 34 533 36 334 37 258.99 34 533 34 063 33 410 34 533 35 800 33 025 26 617 20 262 709 620.6

Table 7 Comparison of computational efficiencies.

Table 6 Comparison of results—strategy 1 and strategy 2. Solution techniques

Minimum operating cost ($)

Strategy 1

ANN-DP [60] ILR [61] IPL-ALH [62] BRCSA

729 326.50 725 996.90 721 352.90 721 182.00

Strategy 1

Strategy 2

SA [8] BRCSA

710 696.00 709 620.60

Strategy 2

Fig. 13. Convergence graph of 26 unit system—strategy 1.

0

Fig. 14. Convergence graph of 26 unit system—strategy 2.

Solution technique

Minimum operating cost ($)

Average computation time (s)

GA PSO BRCSA GA PSO BRCSA

721 310.0 721 208.0 721 182.0 710 652.0 710 557.0 709 620.6

490 511 476 540 563 514

coded CSA tuned fuzzy system. The generating unit data and load profile of 6 unit system is adapted from Ref. [65]. Here the system loss is calculated using B-loss coefficient which is adapted from Ref. [65]. In this case, the real coded CSA is used to tune the fuzzy design variable to maximize the fitness function and to obtain the economic dispatch for 24 h. The parameter tuning is carried out for 6 unit system, and the final combination of parameters that provided the best results are λ = 2.5, initial host nest population = 100 and maximum generation limit = 300. The optimum dispatch of each generator is given in Table 8. For comparison with existing techniques, using the steps given in [40,66,67], the price penalty factor for 6 unit system is calculated as 44.7879 $/kg and it is used to obtain the total cost. The minimum cost so far reported in literature is 55 848.4791 $/h [65] which is 250.6591 $/h higher with respect to the proposed one. For each iteration of the real coded CSA, the fuzzy design variables are tuned for the multi-objective function (cost and emission) which is shown in Fig. 15. The corresponding fitness convergence graph is shown in Fig. 16. From the figures (Figs. 15 and 16), it is clear that the real coded CSA is efficient for tuning the fuzzy design variables and simultaneously determining the best compromise solution using the fitness function equation (27). A real coded CSA tuned fuzzy mechanism picks up the best compromise solution and the results show that the proposed approach is efficient for solving MOOP. Therefore from the Section 8.3, the proposed hybrid approach can be justified for solving the MOUCP which is discussed in Section 8.4.

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Table 8 MOEDP solution – 6 unit system – case 3. Load, MW

700

Method

TOPSIS method [65]

Fuel cost, Fc ($/h) Emission, E (kg/h) Emission cost factor, gc ($/kg) Total cost, ($/h) P1, (MW) P2, (MW) P3, (MW) P4, (MW) P5, (MW) P6, (MW) PL, (MW) Total saving, ($/h)

37 136.3339 36 743.38 417.7938 420.97 44.7879 44.7879 55 848.4791 55 597.82 75.1270 63.5900 75.5840 62.0078 110.0420 112.2560 113.3860 116.2810 165.3620 174.8108 160.3970 171.0544 15.8980 16.2618 TOPSIS method—Proposed method = 250.6591

Proposed method

Fig. 15. Variation of fuzzy design boundaries for Fc and E.

reliability of the system is very low (i.e., high EENS value) due to the shortage of spinning reserve availability for the committed units. When emission is alone minimized, neglecting other objectives, the resultant emission is 17 050 ton/h. The minimum emission is obtained with a higher fuel cost. This is due to the low emission characteristic of sophisticated generating units (P15–P22) involving higher fuel cost. Also the minimization of E involves less number of committed units that leads to low reliability level as mentioned earlier. When EENS is alone minimized, neglecting other objectives, the reliability level of the system can be varied by committing or de-committing generating units. Therefore the EENS is relevant only for UCP through committing and de-committing of generating units which involves SR variation. However for solving single objective function for minimizing EENS, EENS cannot be considered for static economic emission dispatch problem, therefore dispatch problem is solved solely for minimizing both cost and emission. When the UCP is solved for minimization of EENS function, the resultant EENS is 0 (i.e., reliability level is high) and the corresponding cost and emission is $32 746.09 and 20 118.81 ton/h, respectively. Here, the minimization of EENS (i.e., maximization of reliability level) involves more number of committed units that leads to higher fuel cost and emission. It is to be noted that the average time taken to get the maximum and minimum value of each objective function is 500 s. The status and dispatch of the generating units is varied based on the objective function of UCP. The above nature of conflicting results does not give a satisfactory UCP solution. Therefore it stresses the need for the formulation of MOUCP to obtain a best compromise solution between the cost, emission and the reliability level. 8.4.2. Multi-objective function (Fc , E and EENS) The different combinations of objective functions are solved and best compromise results are given in Table 9. It is observed, when the EENS function is combined with other objective functions (Fc and E), the fuzzy set theory controls the system reserve more effectively. The variation of UCP solution for different combination of the objectives gives an intelligent decision on the commitment status of the generating units. 9. Conclusion

Fig. 16. Fitness convergence graph—case 3.

8.4. Case 4: 26 unit system The 26 unit system and the reliability data are adapted from Ref. [68]. The generation cost coefficient is adapted from Ref. [60]. Here all three objective functions (i.e., fuel cost, emission and reliability function) are considered for solving MOUCP using the proposed tuned fuzzified BRCSA. As there are 3 objective functions, the total number of possible combinations is 7. Here a single objective UCP is solved directly using BRCSA while MOUCP is solved using tuned fuzzified BRCSA. The optimum dispatch of each generator is given in Table 9 for different objective functions for a load of 2280 MW (One hour scheduling). 8.4.1. Single objective function (Fc , E and EENS) When UCP is solely solved for the minimization of Fc , minimum cost of $28 119 is obtained by neglecting all other objectives. When fuel cost is alone considered, emission is found to be higher and, a downward trend is seen in the reliability level of the system. Since the generating units are committed solely based on the cost, the

This paper has employed the breeding behavior of cuckoos and the characteristics of Levy flights on the constrained optimization problem. The searching behavior of cuckoos for laying an egg in the host nest is modeled and used for solving power generation scheduling problem.

• The real coded CSA when applied to practical EDP with a non-smooth fuel cost function outperforms with the other techniques reported in the literature in obtaining minimum cost solution in less computational time. • A tanh function is proposed to increase the probability of flipping the status of the binary variable thereby improving the performance of the binary CSA when applied to practical UCP. • The tuned fuzzified real coded CSA when applied to practical MOEDP gives an optimum solution which is found to be superior when compared to other available techniques. • The tuned fuzzy based BRCSA has been presented for solving MOUCP. The problem has been formulated as a MOOP with competing fuel cost, emission and reliability objective functions. The fuzzy design boundary values are tuned to eliminate the dependence of entire global preference information of decision maker in a MOUCP and give a single optimal solution. Thereby the difficulty in fixing the reliability limits and

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Table 9 MOUCP solution – 26 unit system – case 4. Objective fun.

1 objective fun.

2 objective fun.

3 objective fun.

Min Fc

Min E

Min EENS

Min Fc , E

Min Fc , EENS

Min E , EENS

Min Fc , E , EENS

Fuel cost, Fc ($/h)

28 119

34 626

32 746.09

30 740

28 680

34 747

30 149.29

Emission, E (ton/h)

23 952

17 050

20 118.81

21 113

23 081

17 197

21 494.19

EENS (MW h)

1.2948

1.1192

0

0.10975

0.30786

0.31109

0.1539

Reserve, (MW)

94

135

512

351

291

290

315

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26

400 400 350 155 155 155 155 76 76 76 76 75.7 68.744 61.556 0 0 0 0 0 0 0 0 0 0 0 0

400 400 350 95.123 79.877 0 0 76 76 76 76 0 0 0 197 197 197 12 12 12 12 12 0 0 0 0

400 400 350 105.9153 90.4151 74.8592 61.8095 76 76 76 76 25 25 25 197 197 0 12 12 0 0 0 0 0 0 0

400 400 350 147.55 130.86 115.54 100.06 76 76 76 76 25 25 25 197 0 0 12 12 12 12 12 0 0 0 0

400 400 350 155 155 155 155 76 76 76 76 53.091 45.765 38.194 68.950 0 0 0 0 0 0 0 0 0 0 0

400 400 350 66.50 54.25 54.25 0 76 76 76 76 0 0 0 197 197 197 12 12 12 12 12 0 0 0 0

400 400 350 154.9994 140.9706 124.8182 109.2108 76 76 76 76 25 25 25 197 0 0 12 12 0 0 0 0 0 0 0

emission constraints are eliminated in the UCP. The limitation (i.e., trade of between the cost, emission, and reliability level of the system) in selecting a feasible solution from a set of population solutions are circumvented by integrating fuzzy set theory and BRCSA (heuristic approach). The feasibility and performance of the proposed methodology is demonstrated on benchmark test systems for single and multi-objective optimization problems. The comparison of the results with other methods reported in the literature shows the superiority of the proposed methodology and its potential for solving non-smooth multi-objective problems in a power system. The method is promising, simple, easy to implement and applicable for any large-scale power systems. Appendix

Xp1 40.2197 33.2677 7.8252 39.6211 33.3055

Xp2 −64.3735 −74.3971 99.8160 −65.7757 −93.4798.

Step 3: Evaluation of fitness for generated nest position. For a single objective function the fuzzy set theory is not used for the evaluation of fitness function. Hence fitness value of each host nest with an egg is calculated using the Eq. (A.4).

= 1 + abs(F );

when F >= 0 when F < 0.

(A.4)

(A.1)

The objective value F and fitness value Fit for the values Xp1 and Xp2 are generated in the step 2 is given below.

(A.2)

F

Sub.To

−100 <= X(1, 1) <= 100 −100 <= X(1, 2) <= 100.

Variables =

Fit = 1/(F + 1);

Single objective problem. Minimization objective function, F = X(1, 1) ∗ X(1, 2) + 2 ∗ X(1, 1) + 3 ∗ X(1, 2) + 6

Step 2: Initial Generation of host nest population. Generate random number for two variables Xpq (Xp1 and Xp2 ) within the upper and lower limit using Eq. (24). (i.e., p = 1, . . . , m and q = 1, . . . , D. i.e., p = 1, . . . , 5 and q = 1, 2).

(A.3)

Solution: Step 1: Initialization of CSA parameter. Initial population of host nest, m = 5 → Tuning variable λ (constant value, 1 ≤ λ ≤ 3) = 1; → Tuning variable Maximum generation limit = 100; → Tuning variable. Given: Number of variables to optimize, D = 2 (i.e., X(1, 1) and X(1, 2)). Upper limit of two variable: [100100]. Lower limit of two variable: [−100 −100].

= 1.0e + 003 ∗ −2.695767544398496 −2.625682121793714 1.102188887642179 −2.718194441146182 −3.321228745645944 Fit = 1.0e + 003 ∗ 2.696767544398496 2.626682121793714 0.000000906463083 2.719194441146182 3.322228745645944. Model calculation for the 1st nest position X(1, 1) = 40.2197

14

K. Chandrasekaran, S.P. Simon / Swarm and Evolutionary Computation 5 (2012) 1–16

X(1, 2) = −64.3735 F = X(1, 1) ∗ X(1, 2) + X(1, 1) ∗ 2 + X(1, 2) ∗ 3 + 6 = (40.2197 ∗ −64.373) + (2 ∗ 40.2197) + (3 ∗ −64.3735) + 6 = −2.695767544398496 ∗ 1.0e + 003 Fit = 1 + abs(F) = 1 + (2.695767544398496 ∗ 1.0e + 003) = 2.696767544398496 ∗ 1.0e + 003.

Step 4: Set generation = 1. Step 5: Modification of nest position to lay an egg using Levy flights random-walk. Modification of nest position is carried out using the Eq. (20). Here, three random values are generated to modify the position of one host nest position.

• First random value (q) represents the choice of the variables to be modified. i.e., one variable has to be selected out of two for modification. (Select either X (1, 1) or X (1, 2).) • Second random value (f ) is used to choose a neighbor’s nest within the population. i.e., the selected variable has to be modified with respect to the neighbor’s nest. • Third random value α controls the laying of an egg within available host nests through Levy flights equation. Model calculation for the modification 1st nest position Host nest 1: it means p = 1

• Generated first random value is 2 i.e., q = 2. It means X (1, 2) is chosen. • Generated second random value is 5 (i.e., f = 5). It means second variable X (1, 2) is modified with respect to X (5, 2). • Generated third random value α is 0.3383. Also from Eqs. (21) and (20), X (1, 2) is modified as follows: X(1, 2) = X(1, 2) + 0.3383 ∗ (X(1, 2) − X(5, 2)) ∗ 0.8940

= (−64.3735) + 0.3383 ∗ (−64.3735 − (−3.4798)) ∗ 0.8940 = −55.5692. NOTE: Here X (1, 2) and X (5, 2) values are taken from the step 2. Also there will not be any change in the first value. First value X (1, 1)will be same as that of step 2. Similarly, the remaining four nest positions are modified with corresponding random values generated for q, f and α and are given below. f = [3 5 1 1]

−0.2895

−0.1434

0.1790].

Therefore a new modified host nests using above generated values are given below Variables =

= 1.0e + 003 ∗ −2.315250645545016 −4.211916973632900 1.773686340174185 −2.723091563683632 −3.490345841109959 Fit = 1.0e + 003 ∗ 2.316250645545016 4.212916973632900 0.000000563479854 2.724091563683632 3.491345841109959. F

Step 7: Evaluation of probability, Prop . Survival of the host nest in the population is determined using Prop using (23). The probability for the Fitness (calculated in step 6) is calculated as follows Prop = 0.594817627320315 1.000000000000000 0.100000120375472 0.681944154764845 0.845851692939811. Model calculation; For 1st value, Fit (1) = 2.316250645545016 ∗ 1.0e + 003 Max (Fit) = 4.212916973632900 ∗ 1.0e + 003 Pro (1) = ((0.9 ∗ 2.316250645545016 ∗ 1e + 3)/ (4.212916973632900 ∗ 1e + 3)) + 0.1 = 0.594817627320315.

Step 8: Generation of new host nest population. Here one random number Pa is generated between 0 and 1 for each of the host nest. If the random generated number Pa ∈ [0, 1] is greater than the Prop , then the alien egg is identified by the host bird. Then the host bird thrown the alien egg away or abandon the nest and cuckoo find a new host’s nest (in new position) using Eq. (24) for laying an egg. Otherwise, the egg grows up and be alive for the next generation based on the fitness function. Model calculation for 1st nest position. Host nest 1: It means p = 1. Generated random Pa is 0.8261. Here Pa is greater than Pro(1), since there is a chance that the host nest 1 may not survive and therefore cuckoo searches for new host nest using Eq. (24). To find a new nest position generate two random numbers (i.e., rand(0, 1) value in the Eq. (24)).

[0.596387125873

q = [1 1 1 2]

α=[0.9632

and Xp2 ) and are given below. (Model calculation to calculate F and Fit is given in step 3)

Xp1 40.2197 55.1779 14.4204 39.6978 33.30558

Xp2

−55.5692 −74.3971 99.8160 −65.7757 −98.1379.

Here, the modified variable (nest position) is shown in bold letters. NOTE: Here all modified values are with in the upper and lower bound. In case, if it violates the maximum limit set that value to its maximum value. Similarly, if it violates the minimum limit set that value to its minimum value. Step 6: Evaluation of fitness for modified nest position. Using the Eqs. (A.1) and (A.4) the objective function F and corresponding fitness Fit are calculated for the above values (Xp1

0.814723686393179].

Hence using Eq. (24), the X (1, 1) and X (1, 2) are calculated as follows X(1, 1) = Xmin,1 +(0.5963) ∗ (Xmax,1 −Xmin,1 ) = −100 + (0.5963) ∗ (200) = 19.2774 X(1, 2) = Xmin,2 +(0.8147) ∗ (Xmax,2 −Xmin,2 ) = −100 + (0.8147) ∗ (200) = 62.9447.

Here the new nest position with an egg is added with the existing host nest positions which are not destroyed and hence the size of the host nest population is increased by one. (i.e., Number of population of host nest = 5 +1). Now a new added host nest with the available host nest population are given below with bold. Variables =

Xp1 40.2197 55.1779 14.4204 39.6978 33.30558 19.2774

Xp2 −55.5692 −74.3971 99.8160 −65.7757 −98.1379 62.9447.

K. Chandrasekaran, S.P. Simon / Swarm and Evolutionary Computation 5 (2012) 1–16

Similarly, for remaining the host nest (p = 2–5) random value Pa is generated which is given below. Pa =

[0.632359246225410 0.778498218867048

0.097540404999410 0.54688151920494].

When p = 3 and 4, the generated random is greater than the probability calculated in step 7. Hence another two additional nest positions are added with the existing host nest positions. Thereby the size of the host nest is increased to 8 (i.e., Number of population of host nest = 5 + 1 + 2). The new populations of the host nest with the newly added host nest variable which are represented as bold are given below. Variables =

Xp1 40.2197 55.1779 14.4204 39.6978 33.30558 19.2774 39.7070 32.0678

Xp2

−55.5692 −74.3971 99.8160 −65.7757 −98.1379 62.9447 −74.9179 −93.4798.

NOTE: Here all modified values are with in the upper and lower bound. In case, if it violates the minimum/maximum limit then set that value to its maximum value. Similarly, if it violates the minimum limit then set that value to its minimum value. Step 9: Evaluation of fitness for the new host nest population. Using the Eqs. (A.1) and (A.4) the objective function F and fitness Fit are calculated for the above values of Xp1 and Xp2 are given below. (Model calculation to calculate F and Fit is given in step 3)

= 1.0e + 003 ∗ −2.315250645545016 −4.211916973632899 1.773686340174185 −2.723091563683632 −3.490345841109959 1.446799861766400 −3.114107566845203 −3.208000524456982 Fit = 1.0e + 003 ∗ 2.316250645545016 4.212916973632900 0.000000563479854 2.724091563683632 3.491345841109959 0.000000690703202 3.115107566845203 3.209000524456982. F

Step 10: Survival of the host nest with cuckoo’s egg. Sort the fitness function Fit of the new population in descending order with the corresponding nest position.

[4212.9 3491.3 3209 3115.1 2724.1 2316.3 0.000691 0.000563

55.1779 33.3055 32.0678 39.7070 39.6978 40.2197 19.2770 14.4204

−74.3971 −98.1379 −93.4798 −74.9179 −65.7757 −55.5699 62.9447 9.8160].

The first highest fitness value of m number of nest positions is picked up for next generation.

[4212.9 3491.3 3209.0 3115.1 2724.1

55.1779 33.3055 32.0678 39.7070 39.6978

−74.3971 −98.1379 −93.4798 −74.9179 −65.7757].

15

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