ECONOMIC DISPATCH SOLUTION USING BIOGEOGRAPHY-BASED OPTIMIZATION

Aniruddha Bhattacharya#1, Pranab Kumar Chattopadhyay*2

#First Electrical Engineering Department, First Jadavpur University Raja S.C.Mallik Road, Kolkata-32, India 1ani_bhatta2004@rediffmail.com

*Second Professor Electrical Engineering Department Second Jadavpur University Raja S.C.Mallik Road, Kolkata-32, India

2pkchattopadhyay@hotmail.com

Abstract— This Paper presents a Biogeography-Based Optimization (BBO) algorithm to solve Non-convex Economic Load Dispatch (ELD) problems of thermal plants in a power system. The Proposed methodology can take care of economic load dispatch problems involving constraints such as valve point loading, ramp Rate limit and Prohibited Operating zone. Biogeography deals with the geographical distribution of biological species. Mathematical models of Biogeography describe how a species arise, migrates from one habitat to another and gets wiped out. This algorithm searches for the global optimum mainly through two steps: Migration and Mutation. The effectiveness of the proposed algorithm has been verified on two different test systems with valve point loading and prohibited operating zone. Considering the quality of the solution obtained this method seems to be a promising alternative approach for solving the ELD problems in practical power system.Keywords— Biogeography-Based Optimization, Economic Load Dispatch, Prohibited Operating zone Ramp Rate limit, Valve point loading

I. INTRODUCTION

Economic Load Dispatch (ELD) seeks the best generation schedule for the generating plants to supply the required demand plus transmission losses with the minimum generation cost. As better solutions result in significant economical benefits, so as to improve the solution quality, a lot of researches have been done in this area to improve the solution quality. Previously a number of calculus-based approaches including Lagrangian Multiplier method have been applied to solve ELD problems. These methods require incremental cost curves to be monotonically increasing/piece-wise linear in nature. But the input output characteristics of modern generating units are highly non-linear in nature, so some approximation is required to meet the requirements of classical dispatch algorithms. Therefore more interests have been focused on the application of artificial intelligence (AI) technology for solution of these problems. Several AI methods, such as Genetic Algorithm; Artificial Neural Networks, Simulated Annealing, Tabu Search, Evolutionary Programming, Particle Swarm Optimization, Ant Colony Optimization, Differential Evolution, Artificial Immune System, Bacteria Foraging Algorithm have been developed and applied successfully to small and large systems to solve

ELD problems in order to find much better results. Very recently, a new optimization concept, based on Biogeography, has been proposed by Dan Simon [1]. Historical background of biogeography is very interesting. Biogeography describes how species migrate from one island to another, how new species arise, and how species become extinct. A habitat is any Island (area) that is geographically isolated from other Islands. The more generic term “habitat” in this paper is used rather than term “island”. Geographical areas that are well suited as residences for biological species are said to have a high habitat suitability index (HSI). Features that correlate with HSI include factors such as rainfall, diversity of vegetation, diversity of topographic features, land area, and temperature. The variables that characterize habitability are called suitability index variables (SIVs). SIVs can be considered the independent variables of the habitat, and HSI can be executed using these variables. Habitats with a high HSI tend to have a large number of species, while those with a low HSI have a small number of species. Migration of some species from one habitat to other habitat is known as emigration process. When some species enters into one habitat from any other outside habitat is known as immigration process. Habitats with a high HSI have a low species immigration rate because they are already nearly saturated with species. Therefore, high HSI habitats are more static in their species distribution than low HSI habitats. By the same token high HSI habitats have a high emigration rate; the large numbers of species on high HSI islands have many opportunities to emigrate to neighbouring habitats. Habitats with a low HSI have a high species immigration rate because of their sparse populations. This immigration of new species to low HSI habitats may raise the HSI of that habitat, because the suitability of a habitat is proportional to its biological diversity.

BBO mainly works based on the two mechanisms. These are Migration and Mutation. Like GAs and PSO, BBO has a way of sharing information between solutions. GA solutions “die” at the end of each generation, while PSO and BBO solutions survive forever. PSO solutions are more likely to clump together in similar groups, while GA and BBO solutions do not necessarily have any built-in tendency to cluster. Again in BBO poor solutions accept a lot of new

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features from good solutions. This addition of new features to low HSI solutions may raise the quality of those solutions.

These versatile properties of this new algorithm encouraged the authors to apply this newly developed algorithm to solve non-convex complex ELD problems. The performance of the proposed method in terms of solution quality and computational efficiency has been compared with RQEA [4] and other methods for a 13 Generator system and with VSHDE [3] for a 15 Generator system with prohibited operating zone.

Section II of the paper provides a brief description and mathematical formulation of different of ELD problems. The Biogeography is discussed in Section III. The BBO approach is described in Section IV. The simulation results are discussed in Section V. The conclusion is drawn in Section VI.

II. ECONOMIC LOAD DISPATCH The general convex ELD problem considers quadratic cost

function along with system power demand and operational limit constraints. The practical Non-convex ELD problem mainly considers valve point loading effects, prohibited operating zones etc. Basic equations and constraints of non-convex ELD problems are given below:

The objective function Ft of basic ELD problem may be written as

2

1 1min( ( )) min( )

m m

t i i i i i i ii i

F F P a bP cP= =

= = + + (1)

Because of valve point effect, the generation cost Fi(Pi) of i-th generator is represented by

2min( ) { ( )}i i i i i i i i i i iF P a bP cP e Sin f P P= + + + × × −

And tMinF is given by min2

1 1

min ( ) { ( )}m m

t i i i i i i i i i i ii i

F F P a bP cP e Sin f P P= =

= = + + + × × − (2)

Where Fi(Pi), the generator cost function; ai, bi ci , ei and fi

are the cost coefficients of the i-th generator; m is the number of committed generators to the operating system.

It is minimized subject to the following constraints

A. Real Power Balance constraint:

10

m

i D Li

P P P−

=

− = (3)

The total transmission network losses PL can be expressed using B-coefficients as

0 001 1 1

m m m

L i ij j i ii j i

P P B P B P B= = =

= + + (4)

B. Generator Capacity Constraints: The power generated by generator shall be within their

lower limit Pimin or upper limit Pi

max.min max

i i iP P P≤ ≤ (5) The objective function of ELD problem with ramp rate

limits and prohibited operating zone is same as mentioned in

(1). Here the objective function is to be minimized subject to the following constraints

C. Real Power Balance constraint: The constraint due to balance between power generated and

the demand plus losses remains same as in (3).

D. Generator Capacity Constraints: The constraint due to operating limits of Pi i.e.,min max

i i iP P P≤ ≤ remains unchanged as given in (5).

E. Ramp Rate Limit Constraints: The power generated Pi in certain interval may not exceed

that of previous interval Pi0 by more than a certain amount URi, the up-ramp limit and neither may it be less than that of the previous interval by more than some amount DRi the down-ramp limit of the i-th generator. These give rise to the following constraints. As generation increases

0i i iP P UR− ≤ (6)

As generation decreases 0i i iP P DR− ≤ (7)

and min max

0 0max( , ) min( , )i i i i i i iP P DR P P P UR− ≤ ≤ + (8)

F. Prohibited operating zone: The prohibited operating zones in the input-output curve of

generator are due to steam valve operation or vibration in a shaft bearing. Normally the best economy is achieved by avoiding operation in areas that are in actual operation. In practical operation, adjustment of the generation output of a unit must avoid operation in the prohibited zones. Mathematically the feasible operating zones of unit can be described as follows:

min

max

,1

, 1 ,

,

; 2,3,.....i

li i i

u li j i i j i

ui n i i

P P PP P P j nP P P

−

≤ ≤≤ ≤ =

≤ ≤ (9)

where, j represents the number of prohibited operating zones of unit i.Pi,j-1 is the (j-1)-th prohibited operating zone ofi-th unit. Pi,j is the j-th prohibited operating zone of i-th unit. Total number of prohibited operating zone of i-th unit is ni.

III. BIOGEOGRAPHY Actually biogeography is nature’s way of distributing

species, and is analogous to general problem solutions. The problem can be in any area of life (engineering, economics, medicine, business, urban planning, sports, etc.), as long as we have a quantifiable measure of the suitability of a given solution. A good solution is analogous to an island with a high Habitat Suitability Index (HSI), and a poor solution represents an island with a low HSI. High HSI solutions resist change more than low HSI solutions. By the same token, high HSI solutions tend to share their features with low HSI solutions. (This does not mean that the features disappear from the high HSI solution; the shared features remain in the high HSI solutions, while at the same time appearing as new features in

the low HSI solutions. This is similar to representatives of a species migrating to a habitat, while other representatives remain in their original habitat.). Poor solutions accept a lot of new features from good solutions. This addition of new features to low HSI solutions may raise the quality of those solutions. We call this new approach to problem solving as biogeography-based optimization (BBO) [1].

Here, Fig. 1 illustrates a model of species abundance in a single habitat. Let us consider the immigration graph of Fig. 1. The maximum possible immigration rate to the habitat is I, which occurs when there are zero species in the habitat. If a habitat has less number of species then much larger amount of species from other habitat can enter into that habitat, so immigration rate is higher at that time. As the number of species increases, the habitat becomes more crowded, and fewer species are able to successfully survive after immigration to the habitat, and the immigration rate decreases. The largest possible number of species that the habitat can support is Smax, at which point the immigration rate becomes zero, because no more species can enter into that habitat after that species count. Now consider the emigration graph .If there are no species in the habitat then there is no species in that habitat that can shift to other habitat, so the emigration rate must be zero. As the number of species increases, the habitat becomes more crowded, more species are able to leave the habitat to explore other possible residences, and the emigration rate increases. The maximum emigration rate is E, which occurs when the habitat contains the largest number of species that it can support that is Smax. The equilibrium number of species is S0, at which point the immigration and emigration rates are equal. In BBO algorithm, calculation of emigration rate and immigration rate is important as these play vital role to select habitats whose SIVs will undergo migration operation.

Fig. 1. Species model of a single habitat

Mathematically the concept of emigration and immigration can be represented by a probabilistic model. Let us, consider the probability sP that the habitat contains exactly S species at t. sP changes from time t to time t t+ Δ as follows:

( ) ( )( ) 1 1 1 11s s s s s s s sP t t P t t t P t P tλ μ λ μ− − + ++Δ = − Δ − Δ + Δ + Δ (10) where sλ and sμ are the immigration and emigration rates

when there are S species in the habitat. This equation holds

because in order to have S species at time ( )t t+ Δ , one of the following conditions must hold:

1) there were S species at time t , and no immigration or emigration occurred between t and t t+ Δ ;

2) there were ( )1S − species at time t , and one species immigrated;

3) there were ( )1S + species at time t , and one species emigrated.

If time tΔ is small enough so that the probability of more than one immigration or emigration can be ignored then taking the limit of equation (10) as 0tΔ → gives the following equation

( )( )( )

1 1.

1 1 1 1 max

1 1 max

S=0

1 S S -1

S=S

s s s s s

s s s s s s ss

s s s s s

P P

P P P

P PP

λ μ μλ μ λ μλ μ λ

+ +

− − + +

− −

− + +

= − + + + ≤ ≤

− + +

(11)

From the straight-line graph of Fig.1, the equation for emigration rate kμ and immigration rate kλ for k number of species can be written as per following way

kEkn

μ = (12)

1kkIn

λ = − (13)

When E I= , k k Eλ μ+ = (14)

IV. BIOGEOGRAPHY-BASED OPTIMIZATION (BBO) BBO mainly works based on the two mechanisms. These

are Migration and Mutation.

A. Migration With probability Pmod, known as Habitat Modification

Probability each solution can be modified based on other solutions. If a given solution Si is selected to be modified, then its immigration rate is used to probabilistically decide whether or not to modify each suitability index variable (SIV) in that solution. After selecting the SIV for modification, emigration rates of other solutions are used to select which solutions among the population set will migrate randomly chosen SIVs to the selected solution Si.

B. Mutation In BBO species count probabilities are used to determine

mutation rates. The probabilities of each species count can be calculated using the differential equation as mentioned in equation (11).

Each population member has an associated probability, which indicates the likelihood that it exists as a solution for a given problem. If the probability of a given solution is very low then that solution likely to mutate to some other solution. Similarly if the probability of some other solution is higher then that solution set has very little chance to mutate. Mutation rate of each set of solution can be calculated in terms of species count probability using the equation

E

I

S0 Smax

Immigration

Emigration μ

Rat

e

Species count

maxmax

1( ) sPm S mP−= (15)

where mmax is a user-defined parameter.

V. SIMULATION RESULTS

A. Simulation Study and Test Results 1) Test Case 1: In this application a power system with 13-

generating units with valve point loading has been considered. The outputs of generator 11 and 12 are fixed at 75 MW and 60 MW respectively (‘Constrained’).Load demand of the system is 2520MW. Power loss has not been considered here. The cost coefficients along with valve point loading coefficient, operating limits of generators are given in [2]. The load demand is 2520 MW. The result obtained from proposed BBO method has been compared with RQEA [4] and other methods. Best solutions are shown in Table I.

2) Test Case 2: A simple system with 15 thermal units is used to demonstrate ability of the proposed approach. The cost coefficients; operating limits of generators; loss coefficients; prohibited operating zone are taken from [3]. The load demand is 2650 MW. Transmission loss has been considered here. The result obtained from the proposed BBO, different Differential Evolution techniques [3] are shown in Table III.

The following BBO parameters have been used after a number of careful experimentation: habitat size=50, Habitat Modification Probability=1, Immigration Probability bounds per gene=[0,1], step size for numerical integration of probabilities=1, maximum immigration and migration rates for each island=1 and Mutation Probability=0.005.

B. Comparative Study Solution Quality: From the results seen in Tables II and IV,

it is seen that the BBO method can obtain lower generation cost than the other mentioned methods. In Test Case 1, BBO obtains generation cost of 24252.9368 $/hour which is less than the solutions obtained by other methods. In Test Case 2, BBO obtains generation cost of 33260.9556 $/hour which is far less than calculated using VSHDE. Tables II and IV also show that the average costs produced by BBO are least compared with other methods which emphasizes its better solution quality.

Computational Efficiency: The BBO approach is also efficient as far as computational time is concerned. Time requirement is quite less and either comparable or better than other mentioned methods.

TABLE IBEST POWER OUTPUT FOR 13 GENERATOR SYSTEM WITH OUT TRANSMISSION

LOSS (PD = 2520 MW, “CONSTRAINED”) Unit Power Output BBO RQEA [4]

P1 (MW) 628.3185 628.3170P2 (MW) 299.1993 299.1991P3 (MW) 299.1993 299.1990P4 (MW) 159.7331 159.7334P5 (MW) 159.7331 159.7331P6 (MW) 159.7331 159.7330P7 (MW) 159.7331 159.7324P8 (MW) 159.7331 159.7329P9 (MW) 159.7331 159.7331P10 (MW) 107.4846 107.4875

P11 (MW) 75.0000 75.0000P12 (MW) 60.0000 60.0000P13 (MW) 92.39968 92.3994

Total Power Output (MW) 2520.00 2520.00Ploss (MW) 0.0 0.0

Total Generation Cost ($/h) 24252.9368 24252.95

TABLE IICOMPARISON AMONG DIFFERENT METHODS AFTER 50 TRIALS

(13-GENERATOR SYSTEM, WITHOUT TRANSMISSION LOSS, CONSTRAINED)Method Generation Cost ($/h) CPU

Time/Iteration (Sec.)Max. Min. Average

BBO 24252.965 24252.9368 24252.9574 0.049RQEA [4] NA 24252.9500 NA NA

TABLE IIIBEST POWER OUTPUT FOR 15 GENERATOR SYSTEM WITH TRANSMISSION LOSS

(PD = 2650 MW) Unit Power Output BBO VSHDE [3]

P1 (MW) 454.5916 454.74P2 (MW) 453.8755 424.96P3 (MW) 129.9757 129.87P4 (MW) 129.9763 129.99P5 (MW) 335 397.36P6 (MW) 457.7919 460.00P7 (MW) 432.8574 464.75P8 (MW) 61.37939 60.00P9 (MW) 25.29442 25.89P10 (MW) 79.30518 20.75P11 (MW) 20.30728 20.00P12 (MW) 79.54713 75.86P13 (MW) 25.00905 25.06P14 (MW) 15.04721 15.13P15 (MW) 15.00043 15.00

Total Power Output (MW) 2714.9584 2719.35Ploss (MW) 64.9584 69.35

Total Generation Cost ($/h) 33260.9556 33282.17

TABLE IV COMPARISON AMONG DIFFERENT METHODS AFTER 50 TRIALS

(15-GENERATOR SYSTEM, WITH TRANSMISSION LOSS)Method Generation Cost ($/h) CPU

Time/Iteration (Sec.)Max. Min. Average

BBO 33261.032 33260.9556 33261.0026 0.052VSHDE [3] NA 33282.17 NA NA

VI. CONCLUSIONS The BBO method has been successfully implemented to

solve different non-convex ELD problem. The BBO algorithm has the ability to find the better quality solution, has better convergence characteristics, computational efficiency. Due to these properties, the BBO in future can be tried to apply in complex Dynamic ELD problems in the search of better quality results.

REFERENCES[1] Dan Simon, “Biogeography-Based Optimization”, IEEE Transaction

on Evolutionary Computation, Vol. 12, No. 6, December 2008. [2] N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionary

programming techniques for economic load dispatch,”IEEE Trans. on Evolutionary Computation,vol.7,No.1,pp. 83–94,Feb. 03.

[3] J.-P. Chiou, “Variable scaling hybrid differential evolution for large scale economic dispatch problems,” Elect. Power Syst. Res., vol. 77, no. 1, pp. 212–218, 2007.

[4] G. S. S. Babu, D. B. Das, and C. Patvardhan, “Real-parameter quantum evolutionary algorithm for economic load dispatch,” IET Gen., Transm., Distrib., vol. 2, no. 1, pp. 22–31, 2008.