Differential Evolution and Bacterial Foraging Optimization Based ...

Differential Evolution and Bacterial Foraging Optimization Based Dynamic Economic Dispatch with Non-smooth Fuel Cost Functions Kanchapogu Vaisakh1, Pillala Praveena1, and Kothapalli Naga Sujatha2 1

Department of Eelctrical Engineering, AU College of Engineering, Andhra University, Visakhapatnam-530003,AP,India 2 Department of Eelctrical and Electronics Engineering, JNTUH College of Engineering, Jagityal-505501,AP,India {vaisakh_k,knagasujatha}@yahoo.co.in, [email protected]

Abstract. The Dynamic economic dispatch (DED) is an optimization problem with an objective to determine the optimal combination of power outputs for all generating units over a certain period of time in order to minimize the total fuel cost while satisfying dynamic operational constraints and load demand in each interval. Recently social foraging behavior of Escherichia coli bacteria has been explored to develop a novel algorithm for distributed optimization and control. The Bacterial Foraging Optimization Algorithm (BFOA) is currently gaining popularity in the community of researchers, for its effectiveness in solving certain difficult real-world optimization problems. This article comes up with a hybrid approach involving Differential Evolution (DE) and BFOA algorithm for solving the DED problem of generating units considering valve-point effects. The proposed hybrid algorithm has been extensively compared with the classical approach and those reported in the literature. The new method is shown to be statistically significantly better on two test systems consisting of five and ten generating units.

1

Introduction

Dynamic economic dispatch is an extension of the conventional economic dispatch problem used to determine the optimal generation schedule of on-line generators, so as to meet the predicted load demand over certain period of time at minimum operating cost under various system and operational constraints. Due to the ramp-rate constraints of a generator, the operational decision at hour t may affect the operational decision at a later hour. For a power system with binding ramp-rate limits, these limits must be properly modeled in production simulation. The DED is not only the most accurate formulation of the economic dispatch problem but also the most difficult dynamic optimization problem. In the literature, DED problems have been addressed with convex cost functions [1]–[3]. However, in reality, large steam turbines have steam admission valves, which contribute non-convexity in the fuel cost function of the generating units [4]-[6]. B.K. Panigrahi et al. (Eds.): SEMCCO 2013, Part II, LNCS 8298, pp. 583–594, 2013. © Springer International Publishing Switzerland 2013

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K. Vaisakh, P. Praveena, and K.N. Sujatha

Accurate modeling of DED problem will be improved when the valve point loadings in the generating units are taken into account. Furthermore, they may generate multiple local optimum points in the solution space. Previous efforts on solving DED problem have employed various mathematical programming methods and optimization techniques. Traditional methods like gradient projection method [1], Lagrangian relaxation [7], dynamic programming and so on, when used to solve DED problem, suffer from myopia for nonlinear, discontinuous search spaces, leading them to a less desirable performance and these methods often use approximations to limit complexity. The stochastic search algorithms such as genetic algorithm (GA) [4],[8], evolutionary programming (EP) [5],[9],[10], simulated annealing (SA) [11], and particle swarm optimization (PSO) [6] may prove to be very effective in solving nonlinear ED problems without any restriction on the shape of the cost curves. They often provide a fast, reasonable nearly global optimal solution. The setting of control parameters of the SA algorithm is a difficult task and convergence speed is slow when applied to a real system. Though the GA methods have been employed successfully to solve complex optimization problems, recent research has identified some deficiencies in GA performance. This degradation in efficiency is apparent in applications with highly epistatic objective functions. Moreover, the premature convergence of GA degrades its performance and reduces its search capability that leads to a higher probability toward obtaining a local optimum [12]. EP seems to be a good method to solve optimization problems, when applied to problems consisting of more number of local optima the solutions obtained from EP method is just near global optimum one. Also GA and EP take long simulation time in order to obtain solution for such problems. All these methods use probabilistic rules to update their candidates positions in the solution space.. Recently, SA [13], hybrid EP-SQP [14], DGPSO [15] and hybrid PSO-SQP [16] methods are proposed to solve dynamic economic dispatch problem with nonsmooth fuel cost functions. These hybrid methods utilize local searching property of Sequential quadratic programming (SQP) along with stochastic optimization techniques to determine the optimal solution of DED problem. Differential Evolution is one of the excellent evolutionary algorithms [17]. DE is a robust statistical method for cost function minimization, which does not make use of a single parameter vector but instead uses a population of equally important vectors. The BFOA is currently gaining popularity in the community of researchers, for its effectiveness in solving certain difficult real-world optimization problems. This article comes up with a hybrid approach involving Particle Swarm Optimization (PSO) and BFOA algorithm for solving the DED problem of generating units considering valve-point effects. The proposed hybrid algorithm has been extensively compared with the classical approach. The new method is shown to be statistically significantly better on two test systems consisting of five and ten generating units. The results obtained through the proposed method are compared with those reported in the literature.

Differential Evolution and Bacterial Foraging Optimization Based DED

2

585

Formulation of DED Problem

The classic DED problem minimizes the following incremental cost function associated to dispatchable units:

Min F =

T

N

 F

it

( Pit ) ($)

(1)

t =1 i =1

where F is the total generating cost over the whole dispatch period, T is the number of intervals in the scheduled horizon, N is the number of generating units, and Fit(Pit) is the fuel cost in terms of its real power output Pit at time t. Taking into account of the valve-point effects, the fuel cost function of ith thermal generating unit is expressed as the sum of a quadratic and a sinusoidal function in the following form

Fit (Pit ) = a i Pit2 + bi Pit + c i + ei sin( f i ( Pi min − Pit ))

($/h)

(2)

where ai, bi, and ci are cost coefficients, ei, fi are constants from the valve point effect of the ith generating unit, and Pi is the power output of the ith unit in megawatts. The minimization of the generation cost is subjected to the following equality and inequality constraints: 1) Real power balance constraint N

P

it

− PDt − PLt = 0

(3)

i =1

where t = 1, 2, …, T. PDt is the total power demand at time t and PLt is the transmission power loss at time t in megawatts. PLt is calculated using the B-Matrix loss coefficients and the general form of the loss formula using B-coefficients is

PLt =

N

N

 P B it

ij

Pjt

(4)

i =1 j =1

2) Real power generation limit

Pi min ≤ Pit ≤ Pi max

(5)

where Pimin is the minimum limit, and Pimax is the maximum limit of real power of the ith unit in megawatts. 3) Generating unit ramp rate limits

Pit − Pi (t −1) ≤ URi ,

i = 1, 2 , 3 ,..........., N

Pi (t −1) − Pit ≤ DRi ,

i = 1, 2 , 3 ,............., N

(6)

where URi and DRi are the ramp-up and ramp-down limits of ith unit in megawatts. Thus the constraint of (6) due to the ramp rate constraints is modified as

586


max(Pi min , Pi ( t −1) − DRi ) ≤ Pit ≤ min( Pi max , Pi ( t −1) + URi )

(7)

such that

Pit ,min = max(Pi min , Pi (t −1) − DRi )

and

Pit max = min( Pi max , Pi (t −1) + URi )

(8)

4) Constraint satisfaction technique

3

Brief Overview of BFOA and DE Based Hybrid Algorithm

In this section we briefly outline both the BFOA and the DE algorithms The DE has reportedly outperformed powerful meta-heuristics like genetic algorithm (GA) and particle swarm optimization (PSO) [18]. Practical experiences suggest that the DE may occasionally stop proceeding towards the global optima, while the population has not converged to a local optima or any other point. Occasionally even new individuals may enter the population but the algorithm does not progress by finding any better solutions. This situation is usually referred to as stagnation. The DE also suffers from the problem of premature convergence [19] where the population converges to some local optima of a multimodal objective function loosing its diversity. On the other hand, experiments with several benchmark functions reveal that the BFOA possesses a poor convergence behavior over multimodal and rough fitness landscapes as compared to other bio-inspired optimization techniques like GA, PSO etc. [20]. Its performance is heavily affected with the growth of search space dimensionality. Previously to improve the performance of the DE, some attempts have been made to hybridize it with a few local search techniques, and metaheuristics like PSO [19-21]. Recently in 2007 Kim et al. developed a hybrid approach involving the GA and the BFOA for function optimization. Their algorithm outperformed both the GA and the BFOA over several numerical benchmarks and a practical PID tuner design problem. In the present work following the same train of thought, we have incorporated an adaptive chemotactic step borrowed from the realm of the BFOA into the DE. The computational chemotaxis in the BFOA serves as a stochastic gradient descent based local search. It was seen to greatly improvise the convergence characteristics of the classical DE. The resulting hybrid algorithm is referred here as the CDE (Chemotactic Differential Evolution). In the CDE, each trial solution vector first undergoes an adaptive computational chemotaxis. The trial solution is visualized as an E.coli bacterium. During the process of chemotaxis, bacterium in proximity of venomous substance takes larger chemotactic step to move towards the nutrient substances. Before each movement, it is ensured that bacterium moves in the direction of increasing nutrient substance concentration, i.e., region with smaller objective function value. After this, it is subjected to the DE mutation. For the trial solution vector in population three vectors, other than the previous one, are selected. One of the three


587

vectors is added with scaled difference of the remaining two. The vector thus produced probabilistically interchanges its components with the original vector (just like genes of two chromosomes). Offspring vector replaces the original one if the objective function value is smaller for it. The process is repeated several times over the entire population in order to obtain the optimal solution. The brief pseudo-code of the algorithm has been provided below: The CDE (Chemotactic DE) Algorithm Initialize parameters S , N C , , N S , ,

C (i )(i = 1,2...N ), F , CR .

Where, S: The number of bacteria in the population, D: dimension, Nc: no. of chemotactic steps, C(i) : the size of the step taken in the random direction specified by the tumble. F: scale factor for DE type mutation CR: crossover Rate. Set j = 0; t = 0; Chemotaxis loop: j = j + 1; Differential evolution mutation loop: t = t + 1; θ (i, j , t ) denotes the position of the i-th bacterium in the j-th chemotactic and t-th differential evolution loop. for i = 1, 2, . . . , S, a chemotactic step is taken for i-th bacterium. (a) Chemotaxis loop: (i)

(ii) (iii)

Value of the objective function J(i, j, t) is computed where J(i, j, t) symbolizes value of objective function at j-th chemotaxis cycle for i-th bacterium at t-th DE mutation step. Jlast = J(i, j, t) we store this value of objective function for comparison with values of an objective function yet to be obtained in future. Tumble: generate a random vector

Δ(i )∈ ℜ D with each element

Δ m (i ) = 1, 2, ... , D is a random number on [-1, 1]. (iv)

θ (i, j + 1, t ) = ω.θ (i, j , t ) + C (i ).(Δ(i ) / Δ(i ).ΔT (i ) ) . Where ω = inertia factor which is generally equals to 1 but becomes

Move:

0.8 if the function has an optimal value close to 0. C(i) = step size for k-th

(v) (vi)

bacterium

=

(( J (i, j , t )) − 20) /(( J (i, j , t )) + 300) Step size is made an increasing function of objective function value to have a feedback arrangement. J(i, j, t) is computed. Swim: We consider here only i-th bacterium is moving and others are not moving. 1/ 3

1/ 3

588


Now let m = 0; while m < Ns (no of steps less than max limit). Let m = m + 1; If J (i, j , t ) < J last (if going better)

J last = J (i, j , t ). And let, θ (i,

j + 1, t ) = ω.θ (i, j , t ) + C (i ).(Δ(i ) / Δ(i ).ΔT (i ) ) .

Else, m = Ns (end of while loop); for i = 1, 2, . . . , S, a differential evolution mutation step is taken for i-th bacterium. (b) Differential Evolution Mutation Loop: For each θ (i, j + 1, t ) trial solution vector we choose randomly three other distinct vectors from the current population namely θ (l ) + F .(θ (m) − θ (n)), such that i ≠ l ≠ m ≠ n

(i)

(ii) (iii)

V (i, j + 1, t ) = θ (l ) + F .(θ (m) − θ (n)) ,

where, V(i, j + 1, t) is the donor vector corresponding to θ (i, j + 1, t ) . Then the donor and the target vector interchange components probabilistically to yield a trial vector U(i, j + 1, t) following:

Up (i, j +1,t) =Vp (i, j +1,t) If (randp (0,1) ≤ CR) or ( p = rn(i))

θ p (i, j + 1, t ) If (rand p (0,1) > CR ) or ( p ≠ rn(i ))

for p-th dimension.

Where rand p (0,1) ∈ [0,1] is the p-th evaluation of a uniform random number generator. rn(i ) ∈ {1, 2, ... , D} is a randomly chosen index which ensures that

U (i, j + 1, t ) gets at least one component from V(i, j + 1, t). (iv) (v)

J(i, j + 1, t) is computed for trial vector. If

(U (i, j + 1, t )) < J (θ (i, j + 1, t )),θ (i, j + 1, t + 1) = U (i, j + 1, t) Original vector is replaced by offspring if value of objective function for it is smaller. If j < Nc, start another chemotaxis loop.

4

Simulation Results and Discussion

A DE and BFA algorithm for the DED problem described above has been applied to five-unit and ten-unit systems with non-smooth fuel cost function to demonstrate the performance of the proposed method. The simulations were carried out on a PC with Pentium IV 3.1-GHZ processor. The software is developed using the MATLAB 7.1. The number of trials have been conducted with changes in the size of population, number of generations, and number of trials per iteration in order to obtain the best values to achieve the overall minimum cost of generation. The best solution obtained through the proposed method is compared to those reported in the recent literature.


4.1

589

Classical Method

The classic DED problem minimizes the following incremental cost function associated to dispatchable units: Example-1: 5–unit system: The cost coefficients, generation limits, load demand in each interval and ramp-rate limits of five-unit sample system with valve-point loading is taken from Ref. [13]. The scheduling time horizon is one day divided into 24 intervals. The transmission losses are calculated using Bcoefficient loss formula. The optimal dispatch of real power for the given scheduling horizon using the proposed method has been obtained. The best total production cost obtained using classical method is $51119.9. The cost of generation and power loss during 24 time periods are shown in Figs.1 and 2 respectively. The sum of total generating power in each interval satisfies the load demand plus transmission losses. Cost curves

(5-unit system) Classical

3000 2500 2000 Cost($/h)

1500 1000 500 0 1

3

5

7

9

11 13

15 17 19

21 23

hours

Fig. 1. Cost curve fir 5-Unit System

Power Loss for 5-unit system 12 Power Loss

10 8 6 4 2 0 1

3

5

7

9

11

13

15

17

19

hours

Fig. 2. Power Loss for 5-Unit System

21

23

590


Example-2: 10 – Unit System: In this example, the DED problem of the 10-unit system is solved by the proposed method by neglecting transmission losses in order to compare the results of the proposed method with hybrid methods such as Hybrid EPSQP, Deterministically guided PSO and Hybrid PSO-SQP algorithms reported in literature [14], [15], and [16]. The load demand of the system was divided by 24 intervals. The system data for ten-unit sample system is taken from the Ref. [14]. Transmission losses have been ignored for the sake of comparison of results with those reported in literature. The cost of generation during 24 time periods is shown in Figs.3. Cost curve for 10-unit system 60000 50000 cost

40000 30000 20000 10000 0 1

3

5

7

9

11

13

15

17

19

21

23

hours

Fig. 3. Cost curve fir 10-Unit System

4.2

DE and BFO Method Based Hybrid Algorithm

The DED problem minimizes the following incremental cost function associated to dispatchable units: Example-1: 5–Unit System: The cost coefficients, generation limits, load demand in each interval and ramp-rate limits of five-unit sample system with valve-point loading are given in Appendix, which is taken from Ref. [13]. The scheduling time horizon is one day divided into 24 intervals. The transmission losses are calculated using Bcoefficient loss formula. The results of the proposed method are compared with that of the simulated annealing (SA) method [13] and are given in Table 1. The comparison of cost of generation and power loss during 24 time periods are shown in Figs.4 and 5 respectively, The optimal dispatch of real power for the given scheduling horizon using hybrid method is obtained and compared with the results reported in the literature and is given in Table 2. The sum of total generating power in each interval satisfies the load demand plus transmission losses.

Differential Evolution and Bacterial Foraging Optimization Based DED Cost curves for different methods (5-unit system)

591

classical DE-BFO

3000 2500 Cost($/h)

2000 1500 1000 500 0 1

3

5

7

9

11 13

15 17 19

21 23

hours

Fig. 4. Comparison of cost with classical and hybrid method for 5-Unit System Power Loss in different methods (5-unit system)

classical DE-BFO

12 10 Power 8 Loss 6 4 2 0 1

3

5

7

9

11

13

15

17

19

21

23

hours

Fig. 5. Comparison of Power Loss with classical and hybrid method for 5-Unit System Table 1. Best Cost of Generation for 5-unit system using hybrid and SA method METHOD

TOTAL FUEL COST(DOLLARS/24H)

CLASSICAL DE-BFOA SA METHOD

51119.9 46013.0 47356.0

Example-2: 10 – Unit System: In this example, the DED problem of the 10-unit system is solved by the proposed method by neglecting transmission losses in order to compare the results of the improved DE method with hybrid methods such as Hybrid EP-SQP, Deterministically guided PSO and Hybrid PSO-SQP algorithms reported in literature [14], [15], & [16]. The load demand of the system was divided by 24 intervals. The system data for ten-unit sample system is taken from the Ref. [14]. Transmission losses have been ignored for the sake of comparison of results with those reported in literature. The convergence characteristics of maximum fitness and cost of generation for different trials for 10-unit system are shown in Figs.6 and 7 respectively. The comparison of cost of generation during 24 time periods is shown in Fig.8. The comparison of cost of optimum scheduling of generating units for 24 hours using proposed method and the methods reported in the literature is given in Table 2.

592

K. Vaisakh, P. Praveena, and K.N. Sujatha Conv e rge nce characte rstics for diffe re nt trials (10-unit syste m) 9.75E-07 9.70E-07 9.65E-07 trial1 trial2 trial3

Total Fit

9.60E-07 9.55E-07

trial4 trial5

9.50E-07 9.45E-07 9.40E-07 9.35E-07 1

176 351

526

701 876 1051 1226 1401 iterations

Fig. 6. Power loss characteristics for different trails for 10- unit system Convergence characterstics for different trials (10-unit system) trial1 trial2 trial3 trial4 trial5

1070000 1065000 1060000

Total Cost

1055000 1050000 1045000 1040000 1035000 1030000 1025000 1

164

327

490

653

816

979 1142 1305 1468

ite rations

Fig. 7. Generation Cost characteristics for different trails for 10- unit system

Cost curve for different methods (10-unit system) classical DE-BFO

60000 50000 cost 40000 30000 20000 1

3

5

7

9

11

13 15

17

19

21 23

hours

Fig. 8. Comparison of cost of generation with classical and hybrid method for 10 unit system


593

Table 2. Comparison of Best Cost of Generation for 10-unit system

5

METHOD

TOTAL FUEL COST($/24H)

DE-BFOA HYBRID EP-SQP[14] DGPSO[15]

1028800 1031746 1028835

Conclusions

In this paper an hybrid method based on DE and BFA algorithm by combining the DE based mutation operator with bacterial chemotaxis for determination of optimal solution for DED problem with the generator constraints has been presented. The presented scheme attempts to make a judicious use of exploration and exploitation abilities of the search space and therefore likely to avoid false and premature convergence. The feasibility of the proposed method was demonstrated with five and ten-unit sample systems. The test results reveals that the optimal dispatch solution obtained through the DE-BFA lead to less operating cost than that found by other methods, which shows the capability of the algorithm to determine the global or near global solution for DED problem. The proposed approach outperforms SA, hybrid EP-SQP, DGPSO and PSO-SQP methods for DED problems in terms of quality of solution with better performance

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