Electric Power Systems Research 77 (2007) 1381–1389

A clonal algorithm to solve economic load dispatch B.K. Panigrahi a,∗ , Salik R. Yadav b , Shubham Agrawal b , M.K. Tiwari c a

b

Department of Electrical Engineering, Indian Institute Technology, Delhi 110016, India Department of Manufacturing Engineering, National Institute of Foundry and Forge Technology, Ranchi 834003, India c Research Promotion Cell, Department of Forge Technology, NIFFT, Ranchi, India Received 24 May 2006; received in revised form 6 October 2006; accepted 16 October 2006 Available online 20 November 2006

Abstract This paper presents a novel optimization approach to constrained economic load dispatch (ELD) problem using artificial immune system (AIS). The approach utilizes the clonal selection principle and evolutionary approach wherein cloning of antibodies is performed followed by hypermutation. The proposed methodology easily takes care of transmission losses, dynamic operation constraints (ramp rate limits) and prohibited zones and also accounts for non-smoothness of cost function arising due to the use of multiple fuels. Simulations were performed over various systems with different number of generating units and comparisons are performed with other prevalent approaches. The findings affirmed the robustness, fast convergence and proficiency of proposed methodology over other existing techniques. © 2006 Elsevier B.V. All rights reserved. Keywords: Artificial immune system; Clonal selection algorithm; Prohibited operating zones; Ramp rate

1. Introduction Among different issues in power system operation, economic load dispatch (ELD) problem lies at the kernel [1,2]. Essentially, ELD problem is a constrained optimization problem. Over the years, many efforts have been made to solve the problem, incorporating different kinds of constraints or multiple objectives, through various mathematical programming and optimization techniques. The conventional methods include lambda iteration method [3,4], base point and participation factors method [3,4], gradient method [3,5], etc. Among these methods, lambda iteration is most common one and, owing to its ease of implementation, has been applied through various software packages to solve ELD problems. But for effective implementation of this method, the formulation needs to be continuous [14]. Consequently, this method is accompanied by its inability to solve discontinuous ELD problems taking into account the prohibited operating zones. In addition, they have oscillatory problems in large scale and mixed- generating unit systems leading to high computation time. An ELD problem with valve point loading has

∗

Corresponding author. Tel.: +91 11 26591078; fax: +91 11 26591078. E-mail addresses: [email protected] (B.K. Panigrahi), [email protected] (S.R. Yadav), [email protected] (S. Agrawal). 0378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2006.10.007

also been solved by dynamic programming (DP) [6,7]. Though promising results are obtained in small sized power systems while solving it with DP, it unnecessarily raises the length of solution procedure resulting in its vulnerability to solve large size ELD problems in stipulated time frames. Moreover, evolutionary and behavioral random search algorithms such as genetic algorithm (GA) [8–10], particle swarm optimization (PSO) [11,29], etc. have previously been implemented on the problem at hand. Infact, genetic algorithms, based on the theory of genetic evolution, due to their parallel search techniques, have attracted much attention in the past, and were successfully implemented to a variety of electrical engineering problems, viz. hydrogenerator governor tuning [12], load flow problems [13], etc. In addition, an integrated parallel GA incorporating ideas from simulated annealing (SA) and tabu search (TS) techniques was also proposed in Ref. [14] utilizing generator’s output power as the encoded parameter. Yalcionoz has used a real-coded representation technique along with arithmetic genetic operators and elitistic selection to yield a quality solution [16]. GA has been deployed to solve ELD with various modifications over the years. In a similar attempt, a unit independent encoding scheme has also been proposed based on equal incremental cost criterion [15]. In spite of its successful implementation, GA does posses some weaknesses leading to longer computation time and less guaranteed convergence, particularly

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in case of epistatic objective function containing highly correlated parameters [17,18]. Moreover, premature convergence of GA is accompanied by a very high probability of entrapment into the local optimum [19]. In order to alleviate the aforementioned difficulties, this paper proposes a new optimization approach, to solve the ELD, inspired by the characteristics of immune system. Immune system (IS) is a very intricate biological system which accounts for resistance of a living body against harmful foreign entities. Artificial immune system (AIS) imitates the immunological ideas to develop some techniques used in various areas of research [20]. It works on the principle of pattern recognition (distinguishing antibody and antigen) and clonal selection principle, whereby clonal selection algorithm (invariably called as AIS) is implemented to accomplish learning and memory acquisition tasks. In IS, receptors present on the antibodies are responsible for antibody–antigen interaction. In these interactions, different antibodies have different affinity towards an antigen and the binding strength is directly proportional to this affinity [20,21]. AIS effectively exploit these interactions and the corresponding affinity by suitably mapping it to fitness (objective function) evaluation, constraint satisfaction or other relevant amenities of operations research. These ideas are further emulated and thereby harnessed into learning, memory and associative retrieval to solve the optimization problems. A more detailed description relating to theoretical and implementational aspects of the proposed approach is provided in later sections. However, despite its potential, AIS approach has seldom been applied to solve ELD problem [28]. This paper solves the economic load dispatch problem using AIS based solution methodology. In order to establish the capability of AIS to optimize smooth as well as non-smooth cost functions, two alternative models are considered. One being a constrained quadratic cost function with generator constraints, power loss and ramp rate limits while the other being an unconstrained non-smooth cost function with multiple fuels. AIS is tested on 3 generators, 6 generators and 15 generators systems in case of smooth cost curves, and for 10 generator system in case of multiple fuels. The results obtained are compared with those of GA, lambda iteration, PSO and others. The proposed methodology emerges out to be a robust optimization technique for solving ELD problem for various curve natures and different power systems. 2. Problem description In a power system, the unit commitment problem has various sub-problems varying from linear programming problems to complex non-linear problems. The concerned problem, i.e., economic load dispatch (ELD) problem is one of the different non-linear programming sub-problems of unit commitment. The ELD problem is about minimizing the fuel cost of generating units for a specific period of operation so as to accomplish optimal generation dispatch among operating units and in return satisfying the system load demand, generator operation constraints with ramp rate limits and prohibited operating zones. Hereby, two alternative models for ELD are considered, viz.

one with smooth cost functions and the other with non-smooth cost function as detailed below. 2.1. ELD formulation with smooth cost function The objective function corresponding to the production cost can be approximated to be a quadratic function of the active power outputs from the generating units. Symbolically, it is represented as Minimize Ftcost

=

NG 

fi (Pi )

(1)

i=1

where fi (Pi ) = ai Pi2 + bi Pi + ci ,

i = 1, 2, 3, . . . , NG

(2)

is the expression for cost function corresponding to ith generating unit and ai , bi and ci are its cost coefficients. Pi is the real power output (MW) of ith generator corresponding to time period t. NG is the number of online generating units to be dispatched. This constrained ELD problem is subjected to a variety of constraints depending upon assumptions and practical implications like power balance constraints, ramp rate limits and prohibited operating zones. These constraints are discussed as under. (1) Power balance constraints or demand constraints: This constraint is based on the principle   of equilibrium between total NG system generation P i=1 i and total system loads (PD ) and losses (PL ). That is, NG 

Pi = PD + PL

(3)

i=1

where PL is obtained using B-coefficients, given by PL =

NG  NG 

Pi Bij Pj

(4)

i=1 j=1

(2) The generator constraints: The output power of each generating unit has a lower and upper bound so that it lies in between these bounds. This constraint is represented by a pair of inequality constraints as follows: Pimin ≤ Pi ≤ Pimax

(5)

where Pimin and Pimax are lower and upper bounds for power outputs of the ith generating unit. (3) The ramp rate limits: One of unpractical assumption that prevailed for simplifying the problem in many of the earlier research is that the adjustments of the power output are instantaneous. However, under practical circumstances ramp rate limit restricts the operating range of all the online units for adjusting the generator operation between two operating periods [10,11]. The generation may increase or decrease with corresponding upper and downward ramp rate limits. So, units are constrained due to these ramp rate limits

B.K. Panigrahi et al. / Electric Power Systems Research 77 (2007) 1381–1389

as mentioned below.

3. Overview of artificial immune systems

If power generation increases,

Pi − Pit−1

If power generation decreases,

Pit−1

≤ URi

(6)

− Pi ≤ DRi

(7)

where Pit−1 is the power generation of unit i at previous hour and URi and DRi are the upper and lower ramp rate limits, respectively. The inclusion of ramp rate limits modifies the generator operation constraints (5) as follows: max(Pimin , URi − Pi ) ≤ Pi ≤ min(Pimax , Pit−1 − DRi ) (8) (4) Prohibited operating zone: The generating units may have certain ranges where operation is restricted on the grounds of physical limitations of machine components or instability, e.g. due to steam valve or vibration in shaft bearings. Consequently, discontinuities are produced in cost curves corresponding to the prohibited operating zones [1,9]. So, there is a quest to avoid operation in these zones in order to economize the production. Symbolically, for a generating unit i, 

Pi ≤ P pz



and Pi ≥ P pz



(9)



where P pz and P pz are the lower and upper are limits of a given prohibited zone for generating unit i. (5) Conflicts handling in constraints: If ever, the maximum or minimum limits of generation of a unit as given by (8) lie in the prohibited zone for that generator, then some modifications are to be made in the upper and lower limits for the generator constraints in order to avoid the conflicts. In case, maximum limit for a generator lies in the prohibited zone, the lower limit of the prohibited zone is taken as the maximum limit of power generation for that particular generator. Similarly, care is taken in case the minimum limit of power generation of a generator lies in the prohibited zone by taking upper limit of the prohibited zone as the lower limit of power generation for that generator. 2.2. ELD formulation with non-smooth cost function In many practical situations, the objective function of an ELD is non-smooth cost function due to use of multiple fuels. With multiple fuels, the objective function is a superposition of piece wise quadratic function. ⎧ ai1 + bi1 Pi + ci1 Pi2 if Pimin ≤ Pi ≤ Pi1 ⎪ ⎪ ⎪ ⎪ ⎨ ai2 + bi2 Pi + ci2 Pi2 if Pi1 ≤ Pi ≤ Pi2 Fi (Pi ) = . (10) ⎪ .. ⎪ ⎪ ⎪ ⎩ ain + bin Pi + cin Pi2 if Pin−1 ≤ Pi ≤ Pimax where aif , bif , cif are cost coefficients of generator ‘i’ for ‘fth’ fuel so that the total cost in this case is given by Ftcost =

NG  i=1

fi (Pi )

1383

(11)

The development of the vertebrates through the evolution of species is also accompanied by the evolution of the immune system of the species. The immune system of vertebrates contains hierarchically arranged molecules, cells and organs in the body which perform the task of defending the living body from foreign attack. It does so by searching the malfunctioning cells from its own body and foreign disease causing elements. Any element whether belonging to body or foreign, that can be recognized by immune system, is called antigen (self antigen and non-self antigen, respectively), the discrimination of which is based on receptor molecules present on the surface of immune cells. Also, there are two immune cells, viz. B-cells and T-cells differing in terms of medium required for the corresponding antibody–antigen interaction with former interacting freely in solution while the latter doing that via accessory cells. In these interactions, the antibodies (cell receptor, effectively) bind to antigens. Thereafter, through the second signal from the accessory cells via T-cells as helper cells, the antigen stimulates the B-cell to proliferate and mature into another kind of cells called plasma cells. Plasma cells are also called terminal cells, as they do not divide further. In proliferation or cell division, clones are generated which are progenies of a single cell [20,21]. The so-called clone generation takes place in order to achieve the state of plasma cells as they are the most active secretors of the antibodies at a larger rate than rate of antibody secretion by the B-cells. When an antibody recognizes an antigen, then the binding strength is directly proportional to the affinity. These biological principles of clone generation, proliferation and maturation are mimicked and incorporated into an algorithm termed the clonal selection algorithm (CLONALG) [20,21], invariably referred as artificial immune system. While implementing artificial immune system (AIS) using clonal selection principle the main steps in the algorithm include [22,23] activation of antibodies; proliferation and differentiation on the encounter of cells with antigens; maturation and diversification of antibody types by carrying out affinity maturation process through random genetic changes; and removing those differentiated immune cells which posses low affinity antigenic receptors. In order to imitate AIS concept in optimization, the affinity concept is transferred to fitness or objective function evaluation and constraint satisfaction. Here, antigens represent constraints and the antibody–antigenic affinity refers to the extent of constraint satisfaction, i.e., higher the satisfaction of constraints more is the affinity. In addition, if two solutions equally satisfy their constraints, one with better value of the corresponding objective attains larger affinity or fitness value. Initially, a population of random solutions is generated which represent a pool of antibodies which undergo proliferation and maturation. The proliferation of antibodies is realized by cloning each member of the initial pool, i.e., copying each of the initial solution depending on their affinity. Moreover, some implications that are really very important from computational point of view include the effects of affinity on the proliferation rate and maturation rate. The proliferation rate is directly proportional to the affinity, i.e., an immune cell with higher affinity gener-

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ates higher number of offsprings. Obviously, higher affinity is accompanied by a smaller maturation rate. The maturation process is carried through random genetic changes (hypermutation) and removing those differentiated immune cells (clones) which posses low affinity antigenic receptors (fitness value). In IS, low affinity antibodies are secreted by immune cells initially, the immune response becomes more effective in secondary and subsequent encounters due to the presence of memory cells associated with initial infections. Similarly in AIS, the process of proliferation, maturation and antibody–antigen interaction is repeated iteratively, with some suitable stopping criteria, thereby imparting learning in each iteration [22,23]. Fairly large repertoire diversity is maintained by incorporating maturation (or hypermutation) [24,25]. Though self-reactive cells are killed after maturation, it has been found that some of B-cells undergo receptor editing occasionally by deleting their low affinity receptors creating newer ones through V(D)J recombination [26]. In computational terms, receptor editing refers to repairing of infeasibility, if any, resulting due to hypermutation which can easily be incorporated in case of large number of constraints. However, this paper does not exploit receptor editing or repairing strategy in order alleviate the run time complexities. 4. Clonal algorithm: an implementation perspective The AIS is implemented to ELD problem utilizing four main features [21]. Firstly, a pool of immune cells or antibodies is generated. This is followed by proliferation which is actually cloning or copying of the parents. Then, maturation of these clones takes place which is analogous to hypermutation. Thereafter, the antibody–antigen interaction is evaluated followed by the elimination of self reacting immune cells or lymphocytes, i.e., individuals with low affinities or fitness values. 4.1. Encoding, initialization and cloning A population of antibodies is initialized using binary strings each encoding a given solution to ELD problem. That is, the power output from each of the generating unit is encoded into a binary form and a string is formed. This paper refers lymphocytes as the antibody and makes no distinction between a B-cell and its antibody. Each binary string is checked for constraint violation and penalized in case of infeasibility with penalty proportional to the extent of constraint violation. Affinity is calculated via fitness or objective values. Each of the antibodies from the initial pool is copied into a fixed number of clones to generate a temporary population of clones. This population of clones is made to undergo maturation process through hypermutation mechanism. The hypermutation is carried out via affinity based hypermutation rate. Larger hypermutation rate is set for lower affinity clones and vice versa. That is, the probability of hypermutation of each clone is inversely to its affinity. This is followed by their affinity evaluation and penalty in case of any constraint violation. A new population of the same size as initial population of the antibodies is selected from the mutated clones and this completes the first iteration. In the next iteration, this fresh population is made to undergo cloning and hyper-

Fig. 1. Flow chart for ELD.

mutation as discussed above and likewise. A similar approach has previously been implemented on a small size ELD problem with smooth and uniform cost function with no constraints [27]. However, this paper adopts a robust AIS based strategy capable to solve ELD problem with smooth as well as non-smooth cost function comprising of a variety of constraints. Fig. 1 shows this procedure in brief. 4.1.1. Parameter selection and algorithm details The number of bits to encode a solution to the ELD problem is set as fifteen. The size of antibody population is assumed 20 and the number of clones generated per antibody is kept dependent on its fitness value. The hypermutation is performed through stochastic genetic changes in which the probability of hypermutation is made dependent on the fitness of a clone. Affinity evaluation is done by converting the cost function (Eqs. (1)–(11)) into fitness function (12), thereby calculating the fitness value required to carry out a proportional selection of mutated clones. The fitness function to evaluate the affinity of any antibody is given by the following expression: ε(x) =

Ftcost

k + ρ(x)

Fig. 2. Convergence property using AIS.

(12)

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Table 1 Best solution of three units system Unit power output

Improved AIS

GA method

Using two-phase neural network

P1 P2 P3

198.7575 77.99225 35.19886

194.26 50.0 79.62

165 113.4 34.0

192.9 85 34 311.9

Total power output

311.9486

323.89

312.45

PL

11.816

24.011

12.45

Total generation cost

3624.44

3737.2

where ε(x) is the fitness function for string x (antibody), Ftcost the cost associated with that string and ρ(x) is the extent of constraint violation of the string. Fig. 3 shows the implementation of the algorithm through a flow chart. The algorithm starts with the random generation of binary strings which are then decoded into real values to check for constraint violation. In case of any constraint violation, the string is randomly generated again, decoded and checked for violation. This process is repeated iteratively until a deliber-

Lambda iteration

11.86

3652.7

3626.7

ate fixed size of population is attained. When the population becomes full then each of the antibodies is evaluated and clones are generated. The number of clones generated per antibody is dependent in our solution methodology on the affinity or fitness value, i.e., larger number of clones is generated for the antibodies with higher fitness value. The mutation rate is not taken uniform but kept inversely proportional to the fitness value of a given clone (binary flip mutation has been utilized with the probability of mutation varying from 0.035 to 0.010). Consequently, clones with higher fitness are made liable to undergo mutation to a lesser extent as compared to those with lower fitness. Thereafter the mutated clones are decoded into their real values followed by the evaluation of corresponding affinity. This is repeated till all the clones from the temporary clonal population are endured to mutation. Finally, tournament selection is done to select same number of mutated clones as there are in initial population. This completes one generation of the clonal selection algorithm. The convergence parameter is set as the situation when the best solutions of each generation cease to change. Thereby, stopping criteria is taken as convergence or the number of cycles (i.e., the one which is achieved first) subjected to a maximum of 100 cycles. 5. Results and discussions 5.1. ELD with smooth cost function considering ramp rate limits and prohibited operating zones The applicability and viability of the aforementioned technique for practical applications has been tested on three different power systems (3, 6 and 15 generator systems). The obtained Table 2 Best solution of six units system

Fig. 3. Implementation of AIS algorithm.

Unit power output

AIS

PSO method

GA method

P1 P2 P3 P4 P5 P6

458.2904 168.0518 262.5175 139.0604 178.3936 69.3416

447.4970 173.3221 263.4745 139.0594 165.4761 87.1280

474.8066 178.6363 262.2089 134.2826 151.9039 74.1812

Total power output

1275.655

1276.01

1276.03

PL

12.655

12.9584

13.0217

Total generation cost

15448

15450

15459

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Table 3 Best solution of 15 units system Unit power output

AIS

PSO method

GA method

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15

441.1587 409.5873 117.2983 131.2577 151.0108 466.2579 423.3678 99.948 110.684 100.2286 32.0573 78.8147 23.5683 40.2581 36.9061

439.1162 407.9727 119.6324 129.9925 151.0681 459.9978 425.5601 98.5699 113.4936 101.1142 33.9116 79.9583 25.0042 41.4140 35.6140

415.3108 359.7206 104.4250 74.9853 380.2844 426.7902 341.3164 124.7867 133.1445 89.2567 60.0572 49.9998 38.7713 41.9425 22.6445

Total power output

2662.04

2662.4

2668.4

PL

32.4075

32.4306

38.2782

Total generation cost

32854

32858

33113

Table 4 Statistical results for solution qualtity Generation cost (US$) for AIS

3 units 6 units 15 units

Maximum

Minimum

Average

Standard deviation

3637.77 15472 32892

3624.44 15448 32854

3631.468 15459.70 32873.25

3.436 6.252 10.8079

results are compared with the reported results of elitist GA method [14], two-phase neural network [27] and PSO [11]. Following subsections deal with the detailed discussion of the obtained results. 5.1.1. Three-unit system This example considers a simple system with three thermal generating units to demonstrate the working of the AIS

approach. The total load demand is 300 MW. The characteristics of the three thermal units are found in Ref. [14]. The best results obtained are reported in Table 1. The cost obtained by the proposed technique is found to be lesser than the existing results while satisfying all the additional constraints like ramp rate limits and prohibited operating zones besides the basic demand and generation limit constraints. 5.1.2. Six-unit system The system contains six thermal generating units. The total load demand on the system is 1263 MW. Parameters of all the thermal units are reported in Ref. [11]. Results obtained using the proposed AIS are listed in Table 2. It can be evidently seen from Table 2 that the technique provided better results compared to GA and comparable results with the PSO. However, the dispatch levels are dissimilar and the power loss is also lesser. 5.1.3. Fifteen-unit system This system comprises of 15 generating units with a load demand of 2630 MW. For parameters and B loss coefficient matrix of the system one is advised to refer Ref. [11]. The results are given in Table 3. The results obtained are significantly better than GA and comparable with PSO, and also satisfy the system constraints thereby validating the heuristics applicability. 5.1.4. Solution quality The convergence tendency of the proposed AIS based solution strategy for all the three sample cases is plotted in Fig. 2. Fig. 2 shows that the technique converges in relatively fewer cycles thereby possessing good convergence property and resulting in low generation cost. To gain further insights into the solution quality 50 trials were performed for all the test cases and the obtained statistical results are reported in Table 4. From the statistical results it is evident that the generation costs obtained by different trials are closer to the best solution thereby validating that the proposed method has excellent solution quality and convergence characteristics.

Table 5 Best solution for non-smooth cost function (demand = 2400 MW) S

U

HM [31]

MHNN [31,32]

AHNN [30]

IEP (pop = 30) [33]

MPSO (par = 30) [30]

AIS

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

1

1 2 3 4

1 1 1 3

193.2 204.1 259.1 234.3

1 1 1 2

192.7 203.8 259.1 195.1

1 1 1 3

189.1 202.0 254.0 233.0

1 1 1 3

190.9 202.3 253.9 233.9

1 1 1 3

189.7 202.3 253.9 233.0

1 1 1 3

189.683 202.40 253.814 233.019

2

5 6 7

1 1 1

249.0 195.5 260.1

1 3 1

248.7 234.2 260.3

1 3 1

241.7 233.0 254.1

1 3 1

243.8 235.0 253.2

1 3 1

241.8 233.0 253.3

1 3 1

241.94 233.063 253.374

3

8 9 10

3 1 1

234.3 325.3 246.3

3 1 1

234.2 324.7 246.8

3 1 1

232.9 320.0 240.3

3 1 1

232.8 317.2 237.0

3 1 1

233.0 320.4 239.4

3 1 1

232.851 320.452 239.404

TP TC

2401.2 488.500

2399.8 487.87

2400 481.700

2400.0 481.779

2400.0 481.723

2400.0 481.723

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Table 6 Best solution for non-smooth cost function (demand = 2500 MW) S

U

HM [31]

MHNN [31,32]

AHNN [30]

IEP (pop = 30) [33]

MPSO (par = 30) [30]

AIS

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

1

1 2 3 4

2 1 1 3

206.6 206.5 265.9 236.0

2 1 1 3

206.1 206.3 265.7 235.7

2 1 1 3

206.0 206.3 265.7 235.7

2 1 1 3

203.1 207.2 266.9 234.6

2 1 1 3

206.5 206.5 265.7 236.0

2 1 1 3

205.88 206.33 266.48 235.79

2

5 6 7

1 3 1

258.2 236.0 269.0

1 3 1

258.2 235.9 269.0

1 3 1

257.9 235.9 269.6

1 3 1

259.9 236.8 270.8

1 3 1

258.0 236.0 268.9

1 3 1

256.87 236.65 269.2

3

8 9 10

3 1 1

236.0 331.6 255.2

3 1 1

235.9 331.2 255.7

3 1 1

235.9 331.4 255.4

3 1 1

234.4 331.4 254.9

3 1 1

235.9 331.5 255.1

3 1 1

235.51 332.23 255.02

TP TC

2501.1 526.700

2499.8 526.13

2500.0 526.230

2500.0 526.304

2500.0 526.239

2500.0 526.24

Table 7 Best solution for non-smooth cost function (demand = 2600 MW) S

U

HM [31]

MHNN [31,32]

AHNN [30]

IEP (pop = 30) [33]

MPSO (par = 30) [30]

AIS

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

1

1 2 3 4

2 1 1 3

216.4 210.9 278.5 239.1

2 1 1 3

215.3 210.6 278.9 238.9

2 1 1 3

215.8 210.7 279.1 239.1

2 1 1 3

213.0 211.3 283.1 239.2

2 1 1 3

216.5 210.9 278.5 239.1

2 1 1 3

216.01 210.77 278.73 239.47

2

5 6 7

1 3 1

275.4 239.1 285.6

1 3 1

275.7 239.1 286.2

1 3 1

276.3 239.1 286.0

1 3 1

279.3 239.5 283.1

1 3 1

275.5 239.1 285.7

1 3 1

275.25 238.55 286.55

3

8 9 10

3 1 1

239.1 343.3 271.9

3 1 1

239.1 343.5 272.6

3 1 1

239.1 342.8 271.9

3 1 1

239.2 340.5 271.9

3 1 1

239.1 343.5 272.0

3 1 1

239.27 343.07 272.32

TP TC

2600.0 574.030

2599.8 574.260

2600.0 574.370

2600.0 574.473

2600.0 574.381

2600.0 574.381

Table 8 Best solution for non-smooth cost function (demand = 2700 MW) S

U

HM [31]

MHNN [31,32]

AHNN [30]

IEP (pop = 30) [33]

MPSO (par = 30) [30]

AIS

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

F

GEN

1

1 2 3 4

2 1 1 3

218.4 211.8 281.0 239.7

2 1 1 3

224.5 215.0 291.8 242.2

2 1 1 3

225.7 215.2 291.8 242.3

2 1 1 3

219.5 211.4 279.7 240.3

2 1 1 3

218.3 211.7 280.7 239.6

2 1 1 3

218.38 211.66 280.54 239.69

2

5 6 7

1 3 1

279.0 239.7 289.0

1 3 1

293.3 242.2 303.1

1 3 1

293.7 242.3 302.8

1 3 1

276.5 239.9 289.0

1 3 1

278.5 239.6 288.6

1 3 1

278.3 239.65 288.57

3

8 9 10

3 3 1

239.7 429.2 275.2

3 1 1

242.2 355.7 289.5

3 1 1

242.3 355.1 288.8

3 3 1

241.3 425.1 277.2

3 3 1

239.6 428.5 274.9

3 3 1

239.84 428.42 274.95

TP TC

2702.2 625.180

2699.7 626.12

2700.0 626.240

2700.0 623.851

2700.0 623.809

2700.0 623.809

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5.2. ELD problem with non-smooth cost functions due to multiple fuels In order to check the performance of AIS on non-smooth cost functions, which poses challenge to any search heuristic, ELD problem comprising of 10 generators with multiple fuels is considered. It reduces the quadratic cost function into piecewise quadratic cost functions. The input parameters and related constraints of the aforementioned problem are detailed in Ref. [30]. Four cases were considered with input demands varying from 2400 to 2700 MW through increments of 100 MW. For simplicity and ease of comparison with the reported results, network losses are not considered for all the cases. The simulations for all the cases were conducted using the same parameter settings as reported earlier. The best results obtained are detailed in Tables 5–8. As evident from Tables 5–8, AIS provided comparable results as compared to MPSO, and outperformed other approaches. Moreover, the obtained solutions satisfied all the constraints, thus proving that AIS could be considered as a viable alternative to solve the complex ELD problems. 6. Conclusions The paper has employed AIS Algorithm on the constrained economic load dispatch problem. Practical generator operation is modeled using several non-linear characteristics like ramp rate limits, prohibited operating zones and multiple fuels. The proposed approach has produced results comparable or better than those generated by other algorithms and the solutions obtained have superior solution quality and good convergence characteristics. The results obtained for both the cases, i.e., when ramp rate limit and prohibited operating zones were considered as well as when non-smooth cost function is considered, were always comparable or better that the earlier best reported results. From this limited comparative study, it can be concluded that the artificial immune system can be effectively used to solve smooth as well as non-smooth constrained ELD problems. In future, efforts will be made to incorporate more realistic constraints to the problem structure and the practical large sized problems would be attempted by the proposed methodology. Another important aspect of the dispatching algorithm is to look into the frequency control problem when the load is increasing and decreasing. Further research is to be carried out to incorporate the load changing scenario into the dispatch algorithm using the proposed approach. References [1] B.H. Choudhary, S. Rahman, A review of recent advances in economic dispatch, IEEE Trans. Power Syst. 5 (November (4)) (1990) 1248– 1259. [2] H.H. Happ, Optimal power dispatch-a comprehensive survey, IEEE Trans. Power Apparatus Syst. PAS-96 (May/June) (1971) 841–854. [3] A.J. Wood, B.F. Wollenberg, Power Generation, Operation and Control, John Wiley & Sons, New York, 1984. [4] C.L. Chen, S.C. Wang, Branch and bound scheduling for thermal generating units, IEEE Trans. Energy Convers. 8 (June (2)) (1993) 184–189.

[5] K.Y. Lee, et al., Fuel cost minimization for both real and reactive power dispatches, in: IEE Proc. C, Gener. Transm. Distrib., vol. 131, no. 3, 1984, pp. 85–93. [6] A. Bakirtzis, V. Petridis, S. Kazarlis, Genetic algorithm solution to the economic dispatch problem, in: Proc. Inst. Elect. Eng. Gen., Transm. Distrib., vol. 141, no. 4, July, 1994, pp. 377–382. [7] F.N. Lee, A.M. Breipohl, Reserve constrained economic dispatch with prohibited operating zones, IEEE Trans. Power Syst. 8 (February) (1993) 246–254. [8] G.B. Sheble, K. Brittig, Refined genetic algorithm-economic dispatch example, IEEE Paper 94 WM 199-0 PWRS, presented at the IEEE/PES 1994 Winter Meeting. [9] D.C. Walters, G.B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst. 8 (August (3)) (1993) 1325–1332. [10] A.A. Ma El-Keib, R.E. Smith, A genetic algorithm-based approach to economic dispatch of power systems, in: IEEE Conference, 1994. [11] G. Zwe-Lee, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Trans. Power Syst. 18 (August (3)) (2003) 1187–1195. [12] J.E. Lansbeny, L. Wozniak, Adaptive hydrogenerator governor tuning with a genetic algorithm, IEEE Paper 93 WM 136-2 EC, presented at the IEEERES 1993 Winter Meeting. [13] X. Yin, N. Getmay, Investigations on solving the load flow problem by genetic algorithm, EPSR 22 (1991) 151–163. [14] P.H. Chen, H.C. Chang, Large scale economic dispatch by genetic algorithm, IEEE Trans. Power Syst. 10 (4) (1995) 1919–1926. [15] C.C. Fung, S.Y. Chow, K.P. Wong, Solving the economic dispatch problem with an integrated parallel genetic algorithm, in: Proc. Power Con. Int. Conf., vol. 3, 2000, pp. 1257–1262. [16] T. Yalcionoz, H. Altun, M. Uzam, Economic dispatch solution using a genetic algorithm based on arithmetic crossover, in: Proc. IEEE Proto Power Tech. Conf., Proto, Portugal, September, 2001. [17] D.B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, second ed., IEEE Press, Piscataway, NJ, 2000. [18] R.C. Eberhart, Y. Shi, Comparison between genetic algorithms and particle swarm optimization, in: Proc. IEEE Int. Conf. Evol. Comput., May, 1998, pp. 611–616. [19] J.G. Damousis, A.G. Unkirlzis, P.S. Dokopoulos, Network-constrained economic dispatch using real-coded genetic algorithm, IEEE Trans. Power Syst. 8 (November (4)) (1995) 198–205. [20] L. N. de Castro, F. J. Von Zuben, Artificial Immune Systems: Part II—A Survey of Applications. FEEC/Univ. Campinas, Campinas, Brazil. Online available: http://www.dca.fee.unicamp.br/∼lnunes/immune.html, 2000. [21] L.N. de Castro, F.J. Zuben, Learning and optimization using through the clonal selection principle, IEEE Trans. Power Syst. 6 (3) (2002) 239– 251. [22] F.M. Burnet, Clonal selection and after, in: G.I. Bell, A.S. Perelson, G.H. Pimbley Jr. (Eds.), Theoretical Immunology, Marcel Dekker, New York, 1978, pp. 63–85. [23] F.M. Burnet, The Clonal Selection Theory of Acquired Immunity, Cambridge University Press, Cambridge, UK, 1959. [24] C. Berek, M. Ziegner, The maturation of the immune response, Immun. Today 4 (8) (1993) 400–402. [25] A.J.T. George, D. Gray, Receptor editing during affinity maturation, Immun. Today 20 (4) (1999) 196. [26] S. Tonegawa, Somatic generation of antibody diversity, Nature 302 (April (14)) (1983) 575–581. [27] R. Naresh, J. Dubey, J. Sharma, Two phase neural network based modeling framework of constrained economic load dispatch, IEE Proc. Gener. Transm. Distrib., vol. 151, no. 3 (2004) 373–378. [28] T.K.A. Rahman, Z.M. Yasin, W.N.W. Abdullah, Artificial immune based for solving economic dispatch in power systems, in: IEEE National Power and Energy Conference Proceeding, Malaysia, 2004, pp. 31–35. [29] J.B. Park, K.S. Lee, J.R. Shin, K.Y. Lee, A particle swarm opt-imization for economic dispatch with non-smooth cost function, IEEE Trans. Power Syst. 20 (20) (2005) 34–42.

B.K. Panigrahi et al. / Electric Power Systems Research 77 (2007) 1381–1389 [30] K.Y. Lee, A. Sode-Yome, J.H. Park, Adaptive Hopfield neural network for economic load dispatch, IEEE Trans. Power Syst. 13 (May) (1998) 519–526. [31] C.E. Lin, G.L. Viviani, Hierarchical economic dispatch for piecewise quadratic cot functions, IEEE Trans. Power App. Syst. PAS-103 (June (6)) (1984) 1170–1175. [32] J.H. Park, Y.S. Kim, I.K. Eom, K.Y. Lee, Economic load dispatch for piecewise quadratic cost function using Hopfield neural network, IEEE Trans. Power Syst. 8 (August) (1993) 1030–1038. [33] Y.M. Park, J.R. Won, J.B. Park, A new approach to economic load dispatch based on improved evolutionary programming, Eng. Intell. Syst. Elect. Eng. Commun. 6 (June (2)) (1998) 103–110. B.K. Panigrahi is with the Department of Electrical Engineering, Indian Institute Technology, Delhi 110016, India. His research interest is on power quality monitoring, control of FACTS devices and intelligent optimization techniques applied to power system. Salik R. Yadav is a bachelors student with the Department of Manufacturing Engineering, National Institute of Foundry and Forge Technology, Ranchi 834003,

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India. His research interest is on developing AI techniques for optimization applied to mechanical and electrical systems. Shubham Agrawal is a bachelors student with Department of Manufacturing Engineering, National Institute of Foundry and Forge Technology, Ranchi 834003, India. His research interest is on developing AI techniques for optimization applied to mechanical and electrical systems. M.K. Tiwari has been with NIFFT Ranchi, since 1998 as an assistance professor and professor, respectively. He has contributed more than 100 articles in various leading International Journals of repute and international conferences that include IEEE-Systems Man and Cybernetics-Part A, European Journal of Operation Research (EJOR), International Journal of Production Research (IJPR), Journal of Engineering Manufacture, Part-B, etc. He is reviewing articles for about 25 International Journals and serving as an editorial board member of 5 International Journals including IJPR, Robotics & CIM, IJAMT, etc. He has been pursuing collaborative research with several faculty members working in various universities such as Loughborough University, University of Wisconsin at Madison, University of Missouri Rolla, Rutgers University, Kansas State University, Bath University, Caledonian Business School, University of Windson, University of Hong Kong, etc.