[ieee 2013 international conference on computer communication and informatics (iccci) - coimbatore,...

2013 International Conference on Computer Communication and Informatics (ICCCI -2013), Jan. 04-06, 2013, Coimbatore, INDIA

Security constrained Economic Dispatch Using a Hybrid Fuzzy Genetic Algorithm Approach

Dr. Barnali Goswami Electrical Engineering Department

Assam Engineering College Guwahati, India

[email protected]

Vanita Agrawal Electrical Engineering Department

Royal School of Engineering & Technology Guwahati, India

[email protected]

Abstract— Premature convergence, which occurs due to loss of diversity, is one of the major issues in Genetic Algorithm. The adaptive probabilities of cross-over and mutation can be used to realize the twin goals of maintaining diversity in the population and sustaining the convergence capacity of the Genetic Algorithm. In this work, the Genetic Algorithm operators of cross-over and mutation are varied adaptively using the fuzzy logic approach and it has been used to reach an optimal solution for the Security Constrained Economic Dispatch problem. A comparative study has been made between Simple Genetic Algorithm and Fuzzy Genetic Algorithm on IEEE 6- bus 3-unit and IEEE 30-bus 6-unit systems. Results of comparison with other techniques are presented, showing the better computational efficiency and convergence property of the proposed approach.

Keywords- Genetic Algorithm, Security Constrained Economic Dispatch, Fuzzy Logic

I. INTRODUCTION Genetic Algorithm (GA) is an optimization algorithm and

is suitable for finding solutions to complex non-linear problems quite efficiently and accurately [1]. Though the Simple Genetic Algorithm (SGA) locates the neighborhood of near-optimal solutions, it often gets trapped in the local optima. The main problem of SGA is premature convergence before the true global optimum solution has been found [2].So, to find the global optimum it is necessary that the algorithm searches the unexplored regions of the solution space vigorously. In this work, the GA parameters namely cross-over probability and mutation probability is varied according to the performance of the GA after each generation (iteration). This helps the GA to search for the global optimum solution as is evident from the results.

The Security Constrained Economic Dispatch (SCED) problem has been solved using a number of approaches including classical and artificial intelligence methods [10]. Some authors have tried to incorporate the use of Fuzzy logic in solving SCED problems. Laoufi, Hazzab and Rahli have applied fuzzy logic to control the change in probability of mutation (pm) and probability of cross-over (pc) and population size. They have also applied GA to automatically design and tune the fuzzy knowledgebase system [3]. Herrera and Lozano presented an extensive study of the application of fuzzy logic controllers to adaptive GA’s [4]. They defined two new

operator types which determined the selective pressure in the case of linear ranking mechanism which were adjusted using a Fuzzy Logic Controller.

In this work the pc and pm are varied by the Fuzzy Logic Controller which computes new values of these operators based on the current performance measures of the GA. These performance measures are the diversity measures which are computed using the fitness values in each generation. This hybrid approach gives satisfactory results in locating the global optimum as can be seen from the results.

II. PROBLEM FORMULATION

A. Security Constrained Economic Load Dispatch SCED solution pertains to finding the optimum allocation

of generation amongst the various available generating units so that the cost of generating power is minimized while simultaneously satisfying the total load demand of the system and the voltage and line flow constraints [5].

If there are n generators committed to the system and the load (active power) demand is PD, then the SCED problem can be stated as follows:

Minimize FC (PG) = ∑i=1 n FC (PGi) (1)

Where FC (PG) is the total fuel cost of generation of all the n generators.

The FC (Fuel Cost) of the i th generating unit can be written as a quadratic polynomial.

FC (PGi) = ci +bi *(PGi) + ai *(PGi) 2 (2)

Where suffix i stands for the unit number, FC(PGi) is the fuel cost expressed in some monetary unit of the ith unit, PGi is the ith unit output in KW and ai, bi and ci are the cost co-efficient of the ith generating unit.

Subject to the given constraints:

A. Equality Constraint: The total system generation equals the total load demand (PD) on the system.

∑i=1 n PGi = PD (3)

B. Inequality Constraint:

(i) Generator constraints: Active power output of each

978-1-4673-2907-1/13/$31.00 ©2013 IEEE


generating unit ranges between a certain minimum and maximum limits so as to ensure stable operation.

PGi min ≤ PGi ≤ PGi max (4)

(ii) Security Constraints: The voltage at the buses and the MVA line flow in the transmission lines stays within their maximum and minimum limits.

(a) V i,min ≤ Vi ≤ V i,max (5)

(b) S ij, min ≤ S ij ≤ S ij, max (6)

Where V i represents the voltage at bus i; V i,min and V i,max are the minimum and maximum voltage limits at the bus i; S ij is the line flow between bus i and j; S ij, min and S ij, max are the minimum and maximum limits of line flow (MVA limit).

III. THE HYBRID APPROACH In this section the Genetic Algorithm and the Fuzzy Logic

Controller approach used to solve this problem has been described.

A. Genetic Algorithm The process of Genetic Algorithm (GA) begins by the

random generation of the initial population of possible solutions. Each possible solution is an ‘individual’ which represents a load dispatch schedule. This population then undergoes evolution towards the ‘fitter’ individuals. The evolution is carried out in generations with the help of genetic operators like mutation and cross-over probability [6]. These individuals, modified by the operators then form the new population and the process continues till the maximum number of generations is reached. At the end of the search process, we get the optimum dispatch schedule.

B. Fuzzy Logic Controller Fuzzy Logic was developed by L.A. Zadeh in 1965 to

address the issue of uncertainty and imprecision in real world problems. This ‘imprecision’ or fuzziness is the core of Fuzzy Logic applications. In Power Systems, where the search space or the solution space is highly complex and non-linear, Fuzzy Logic has been applied to solve a variety of problems.

A Fuzzy Logic Controller (FLC) maps a non-linear input data set to a scalar output data. It consists of four main components: Fuzzifier, Rules, Inference Engine and Defuzzifier. Firstly, a crisp set of input data are gathered and converted to a fuzzy set using fuzzy linguistic variables, fuzzy linguistic terms and membership functions. This is known as fuzzification. After this an inference is made based on a set of rules. Lastly, the resulting fuzzy output is mapped to a crisp output using the membership functions, in the defuzzification step.

The process of fuzzy logic is explained in the following algorithm:

1. Define the linguistic variables and terms (initialization).

2. Construct the membership functions (initialization). 3. Construct the Rule base (initialization).

4. Convert crisp input data to fuzzy values using the membership functions (fuzzification).

5. Evaluate the rules in the rule base (inference). 6. Combine the results of each rule (inference). 7. Convert the output data to non-fuzzy values

(defuzzification).

C. GA based on FLC As mentioned earlier, GA’s are quite robust algorithms’

since they provide answers to problems which are otherwise quite difficult to solve. They have a lot of advantages over conventional optimization methods but still suffer from the drawback of ‘premature convergence’. They have a tendency to converge to the local optimum solution instead of a global optimum solution.

To avoid being trapped in local optima GA’s have to be sensitive to the search process. Certain performance measures (diversity measures) from the current ‘state’ of the GA search process are given as inputs to the FLC [4]. The FLC then computes the values of the operators (pc and pm) based on the Rule Base. These values are then used by the GA search process to look for the global optima.

IV. IMPLEMENTATION STEPS In this work an adaptive real coded GA [7] based on FLC

for the solution of SCED is proposed. The FLC uses two diversity measures Gdm1 and Gdm2 as inputs. The outputs of the FLC are: (i) pc: cross-over probability (ii) pm: mutation probability.

A. Diversity Measures Diversity measures used for adaptation of GA parameter

settings are defined through the combination of some of the following measures:

i. Average fitness ( fav) ii. Best fitness (fbest)

iii. Worst fitness (fworst) In the present approach i.e. Fuzzy Genetic Algorithm

(FGA), two diversity measures (Gdm1 and Gdm2) are defined as follows:

Gdm1 = fav / fbest ; Gdm2 = fworst / fbest (7)

Gdm1 and Gdm2 belong to the interval [0 1]. When they are near to one, it indicates all individuals have almost the same value; in this case diversity is the smallest; so to avoid premature convergence, probability of mutation and probability of cross-over must be changed in such a way as to introduce new genetic characteristics but also at the same time reduce the loss of genetic material. So pm must be augmented and pc reduced. In the other case, if Gdm1 and Gdm2 are much less than one then it means that there is a lot of genetic diversity introduced by the mutation operation. To avoid a basically random search, pm must be reduced and pc augmented.


B. Implemented Algorithms The following algorithm is used in this work for solving the

real-coded SCED problem incorporating Fuzzy Logic Control in the GA:

I. Read generator data (a, b & c coefficients), Pmin & Pmax limits, Power demand (PD) and the bus data and the line data. Set the limit for maximum number of generations (NGEN) and the number of populations (NPOP).

II. Generate the initial population of solutions randomly. The individuals in the ‘parent’ solution thus generated are the real power outputs of the committed NG number of generating units.

III. Set the generation counter and the population counter to 1.

IV. Perform Newton Raphson load flow calculations and the line flow calculations for each of the individual solutions. Determine the amount of voltage violation at the buses and the MVA line flow violation in the lines.

V. Calculate the actual fuel cost for each generating unit for a particular individual solution using (2).

VI. Sum up the cost for all the generating units of a particular individual solution.

FC (PG) =∑i=1NG FC (PGi) (8)

Where FC (PG) is the total fuel cost (for each individual solution).

VII. Form the penalty function for each solution.

PF =K1* (LOAD-∑i=1NG PGi) 2 +K2 *∑i=1

Nl (Ii –Imax) 2 +K3 *∑i=1

NB (Vi –Vmax) 2 (9)

Where PF is the penalty function, K1 , K2 & K3 are positive constants (usually a very large number) reflecting the constraints weight, Ii is the line flow (MVA) of i th line, Vi is the voltage at the i th bus, NL is the total number of transmission lines and NB is the total number of buses in the system..

Add this penalty function to the total fuel cost (FC (PG)) to get the new Objective Function.

F (PG) = PF+ FC (PG) (10)

Where F (PG) is the new objective function.

VIII. Calculate the fitness of each individual solution as:

Fit = 1/F (PG) (11)

Where ‘Fit’ is the fitness of each individual solution.

IX. Repeat the steps from step number IV to step number VIII till the fitness of all the individuals in a population have been found out.

X. Find out the value of the fitness of the ‘bestfit’ solution (the solution having maximum fitness) and the fitness of the ‘worstfit’ solution (the solution having least fitness value). Calculate the average

fitness of the entire population of solutions. Calculate the diversity measures as given by (7).

XI. Construct the membership function for the two diversity measures (inputs of FLC) and probability of cross-over (pc) and probability of mutation (pm) values (outputs of FLC).

XII. The Rule Base that describes the relation between inputs and outputs is given in Table I.

TABLE I. RULE BASE

Gdm1 Gdm2 Pc Pm

vs vs m vs vs s s s vs m s s

vs b b s

s vs s s

s s s s

s m vs m

s b m s

m vs s m

m s vs m

m m vs m

m b s s

b vs vs b

b s vs b

b m s m

b b s s

XIII. Defuzzify the outputs of the FLC to get the value of pc

and pm using the centroid method of defuzzification.

XIV. Select the individuals for the mating pool according to their fitness values and then perform cross-over and mutation using pc and pm.

XV. The new population (formed after cross-over and mutation) replaces the old population, and then the generation counter is incremented by 1 and the new population is again subjected to the same process from step number IV, till the maximum number of generations (NGEN) is reached.

XVI. The optimum solution for the security constrained economic load dispatch is then obtained.

V. RESULTS AND DISCUSSIONS The Security Constrained Economic Dispatch problem has

been approached using Simple GA (SGA) and Fuzzy GA (FGA). To prove the effectiveness of the proposed algorithm (FGA), it has been tested on IEEE 6-bus 3-unit system for a load of 400 MW without incorporating security constraints initially and then on IEEE 30-bus 6-unit system incorporating the security constraints. The data has been taken from [8] and [6] respectively.


The convergence of fuel cost is shown in Fig.1 for IEEE 6-bus 3-unit system for a load of 400 MW without incorporating security constraints. Table II presents a comparison of fuel costs with other techniques for a load of 400 MW for IEEE 6 bus 3 unit system.

The objective function is the total fuel cost represented by quadratic cost functions as given by (8). In this work a dynamic penalty function method is used for constraint handling. A penalty proportional to the extent of constraint violation is added to the objective function value if any violation (voltage and line flow) occurs. Penalty methods use a mathematical function that will increase the objective function value for any given constraint violation as given in (9).

In the case of FGA, triangular membership function with height unity has been used to quantify a linguistic term for the two inputs and the two outputs. Four linguistic terms vs (very small), s (small), m (medium) and b (big) have been used. For each linguistic term, there is a triangular fuzzy set that defines its semantic, i.e. its meaning. The database i.e. the linguistic label associated with each input and output is shown in Figs. 3-6.

In a FLC, a Rule Base is constructed to control the output variable. All the inputs and outputs use the same linguistic variables. Based on the linguistic, rules are established. A fuzzy rule is a simple IF-THEN rule with a condition and a conclusion. Table I shows the matrix representation of the fuzzy rules. First two columns contain the inputs gdm1 and gdm2 which are diversity indicators. Next two columns denote the output pc and pm. The interpretation for the first row of the table is that IF gdm1 and gdm2 are very small THEN pc and pm have to be medium and very small respectively. Similarly, other rows are interpreted.

Cross-over probability and mutation probability were bound by rates [0.5 1.0] and [0.001 0.05] respectively.

A population size of 50 is chosen. Population size is kept the same in both the cases for the purpose of comparison. The proposed algorithm has been implemented in the MATLAB environment.

The results obtained have been compared with those obtained from some other techniques in previously published works and it has been found that a better optimal solution, having lower overall minimum cost of generation can be obtained from the proposed methodology (FGA), as can be seen from Table II and IV. In Table III simulation results are tabulated for IEEE 30 Bus system for SCED for a load of 283.4 MW and where P1-P6 are the unit outputs.

The fitness function convergence characteristics curves are plotted for the two cases i.e. SGA and FGA for IEEE 30 Bus system as shown in Fig. 2 for SCED case. It is seen that, the initial fitness values for the two cases are different from each other. This is due to the fact that, the initial population is chosen through a random process. It can be seen that, the fitness function converges smoothly to the optimum value without oscillations. This shows the convergence reliability of the proposed algorithm.

The power system equipment is designed to operate at fixed voltages with allowable variations of 5-10% of the rated values. The voltage profile of one of the initial population solutions for IEEE 30 Bus system is shown in Fig. 7, where the maximum voltage limits of 1.1 pu has been exceeded. Fig 8 shows the voltage profile at the buses without maximum limit violation in the optimum schedule. This proves the efficacy of the proposed methodology (FGA) in solving SCED problem in a reliable manner.

Line flow limit violations in the initial population have been reported. The line flow limit has been violation in one such case is shown in Fig. 9. The problem has been solved using the proposed algorithm (FGA) and the line flows corresponding to optimum schedule are shown in Fig. 10. There is no line flow limit violation in the case corresponding to the optimum schedule. This fact demonstrates that the proposed algorithm is reliable in eliminating the line flow limit violations.

TABLE II. COMPARISON OF FUEL COSTS (RS./HR) FOR ELD WITH OTHER TECHNIQUES WHEN LOAD=400MW FOR IEEE 6 BUS SYSTEM

GA [8] PSO [8] SGA FGA

20828.5 20812.4 20513 20500

TABLE III. SIMULATION RESULTS OF SCED OF IEEE 30 BUS 6 UNIT SYSTEM

Parameters SGA FGA

P1 (MW) 137.0804 137.2411

P2 (MW) 52.4520 57.6718

P3 (MW) 35.3219 27.3223

P4 (MW) 19.3034 12.8398

P5 (MW) 12.8986 23.8060

P6 (MW) 26.3435 24.5189

Total Cost($/hr) 801.2817 795.5012

TABLE IV. COMPARISON OF FUEL COSTS (($/HR) WITH OTHER TECHNIQUES FOR SCED FOR IEEE 30 BUS SYSTEM

[6] [9] SGA FGA

806.42 802.4 801.3 795.5


Figure 1. Convergence of Fuel Cost for IEEE 6 Bus 3 Unit system without Security Constraints.

Figure 2. Convergence of Fuel Cost for IEEE 30 Bus 6 Unit system with Security Constraints.

Figure 3. Triangular membership function for pm

Figure 4. Triangular membership function for pc

Figure 5. Triangular membership function for gdm1

Figure 6. Triangular membership function for gdm2

Figure 7. Voltage profile at the buses without Security Constraints for IEEE 30 Bus


Figure 8. Voltage profile at the buses with Security Constraints for IEEE 30 Bus

Figure 9. Line Flow with violation at line 1 for IEEE 30 Bus

Figure 10. Line Flow without violation for IEEE 30 Bus system

VI. CONCLUSION In this work a real coded GA based on FLC for the solution

of SCED is proposed. The classical optimization techniques use gradient-based information by linearizing the objective function and the system constraints around the operating point. Thus, these methods that make use of derivatives and gradients are not suitable to locate the global optimum for problems like SCED which are highly non-linear. GA’s solve this problem

but suffer from the drawback of premature convergence. In this work, an efficient and simple approach (FGA) has been applied to solve the security constrained economic dispatch problem. The approach has been tested on the IEEE 6-bus 3-generator for Economic Load Dispatch (ELD) and IEEE 30-bus 6-generator systems for SCED. Simulation results demonstrate the ability of the FGA to solve the SCED problem efficiently. As the total fuel cost obtained is less in the case of the FGA, it is clear from the results obtained, that the FGA gives a better optimal solution than the other techniques. The proposed approach is relatively simple, reliable and efficient.

ACKNOWLEDGMENT The second author would like to thank the first author for

her valuable guidance and support and also to the Electrical Engineering department of Assam Engineering College for providing the necessary support.

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