 Open Access
 Total Downloads : 532
 Authors : Suranjana Bharadwaj, Bipul Kumar Talukdar
 Paper ID : IJERTV5IS060063
 Volume & Issue : Volume 05, Issue 06 (June 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS060063
 Published (First Online): 01062016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Economic Load Dispatch of a Three Unit Thermal System : Using Both Analytical Method and Fuzzy Logic
Suranjana Bharadwaj1 Bipul Kumar Talukdar2
Department of Electrical Engineering
Girijananda Chowdhury Institute of Management and Technology Guwahati, India
Abstract This paper gives a theoretical analysis of the economic load dispatch problem by taking a three unit thermal system as the reference. Load allocation to each unit is determined using analytical method keeping in mind that it serves the total system load and satisfies the generation constraints. A MATLAB program is developed which gives the load allocation as output when the system load is varied step by step. Results are shown with the help of both tables and graphs. In the second phase system is again analyzed using a fuzzy controller and a comparative analysis is done in terms of the fuel costs obtained from conventional method and the fuzzy controller. The whole analysis is done without considering the transmission losses.
KeywordsLoad, Demad, Lambda, Fuzzy Controller, Membership Function, Fuel Cost

INTRODUCTION
Economic Load Dispatch (ELD) problem is one of the major concerns for power engineers as it is directly related to the reliability, stability and economy of the system. The prime concern of ELD scheme is to allocate the total system load among all the generators of the system in such a way that the fuel cost is minimized and at the same time the generation constraints are also satisfied.
The total cost of generation can be divided into two parts
i.e. fixed cost and variable cost. The fixed cost comprises largely of the capital investment and some other miscellaneous expenses which do not have any relation with the load demand or the generator loadings. The variable cost mainly comprises of the generator loadings, system losses and purchase or sale of power.
The cost minimization using ELD scheme is thereby refers only to the variable cost. The fixed cost is indifferent of the system generation or load demand.

THE INCREMENTAL FUEL COST
The incremental fuel cost, mostly referred to as lambda() is one of the key terms in economic load dispatch. It is the minimum value of the operating cost that is obtained after equalizing all the partial derivatives of the cost functions of all the plants and also satisfying the load demand.
=dCi/dPi, i=1,2,.n
Where Ci is the cost function of the ith unit, Pi is the generation of the ith unit and n is the total number of units.
III CONSTRAINTS OF ECONOMIC DISPATCH
The first constraint is the equality constraint i.e. the total generation must be equal to the total system load provided losses are neglected.
Pi=Pload, i=1,2,.n
The second constraint is the inequality constraint. It ensures that the load allocated to each generator lies between the minimum and maximum limits of generation of that particular unit.
Pimin Pi Pimax
IV SYSTEM UNDER INVESTIGATION
The system considered here is a three unit thermal system whose cost functions and constraints are as given below:
Unit1:
C1=300+1.6P1+0.005P12 Rs/MW 80 P1250
Unit2:
C2=250+2.02P2+0.001P22 Rs/MW 300 P1 700
Unit3:
C3=160+2.03P3+0.025P32 Rs/MW 60P1 550
The system is solved using analytical method with the help of a MATLAB program which takes load demand as the input and gives out power generation as the output for all the three units while satisfying all the constraints mentioned earlier.
The algorithm for analytical solution is mentioned below:
step1: Start
step2: Input the system load demand step3: Calculate P1, P2, P3,
step4: Check if any of the generation constraints are violated.
If they are within limit then go to step6
step5: If limit is violated for the ith unit then fix it at the margin value i.e. Pi=Pmax and Pload=PloadPi
step6: Again reschedule the remaining units with the new load demand.
step6: Display P1, P2, P3, step7: Stop
V ANALYTICAL SOLUTION METHOD
Using the analytical method load allocations to the units have, incremental fuel cost and total fuel cost been found for different values of load demand(Pd) as input which can be observed from the following table. It is observed that on increasing the load, both incremental cost and fuel cost rises and the pattern has been shown with graphs. All the graphs show an increasing pattern.
TABLE I: RESULTS OF ANALYTICAL METHOD
Pd 
P1 
P2 
P3 
FC(total) 

450 
100 
290 
60 
2.6 
1801.7 
480 
105 
315 
60 
2.65 
1880.45 
500 
108.3 
331.67 
60 
2.68 
1933.78 
550 
116.67 
373.33 
60 
2.76 
2070.03 
600 
125 
415 
60 
2.85 
2210.45 
650 
133 
456 
60 
2.93 
2355.03 
700 
141 
498 
60 
3.01 
2503.78 
720 
145 
515 
60 
3.05 
2564.45 
750 
150 
540 
60 
3.10 
2656.7 
780 
155 
565 
60 
3.15 
2750.45 
800 
158 
581 
60 
3.18 
2813.78 
850 
166.67 
623 
60 
3.26 
2975 
The different responses obtained are as shown below:
Fig1: Generation vs. incremental cost curve for 3 units
Fig2: Generation vs. load demand curve for 3 units
Fig3: Total fuel cost vs. load demand curve for 3 units
Fig4: Total fuel cost vs. incremental cost curve for 3 units

IMPLEMENTATION OF FUZZY LOGIC CONTROLLER
The fuzzy logic controller provides an alternate way to solve the economic dispatch problem. The fuzzy controller implemented here is a two input one output system. The designing and tuning of the controller is done manually on the basis of understanding the behavior of the output in response to the change of input quantities. The input quantities are demand Pd and incremental cost . The output quantity is power generation of respective units. Three separate but similar type of controllers are used for the three units. The input output relation is given by the following rule table:
TABLE II: RULE TABLE FOR P1, P2, P3
Pd /
L
M
H
L
L
L
L
M
L
L
L
H
M
M
M
The terms L, M, H stands for Low, Medium and High respectively. Hence the rules will be depicted as: If Pd is Low and isLow then P is Low. The Membership functions for the Pd, and P1 are shown below. Triangular membership functions are used. For defuzzification, Centroid method has been used.
Fig 5: Membership function for
Fig 6: Membership function for Pd
Fig 7: Membership function for P1

RESPONSES OF FUZZY LOGIC CONTROLLER
The following table lists out the generation scheduling given by the fuzzy logic controller in response to a wide variation of system load and also gives a comparison between the total fuel costs between analytical solution method and fuzzy controller method.
TABLE III: FUZZY CONTROLLER OUTPUT
Pd 
Fuel Cost (Analytical) 
Fuel Cost (Fuzzy) 

450 
2.6 
1801.7 
2750 
480 
2.65 
1880.45 
2750 
500 
2.68 
1933.78 
2750 
550 
2.76 
2070.03 
2760 
600 
2.85 
2210.45 
2750 
650 
2.93 
2355.03 
2750.05 
700 
3.01 
2503.78 
2750 
720 
3.05 
2564.45 
2823.6 
750 
3.10 
2656.7 
3035.6 
780 
3.15 
2750.45 
3180.9 
800 
3.18 
2813.78 
3170.7 
850 
3.26 
2975 
3409.8 
Fig 8: Comparison Between Fuel Costs
V CONCLUSION
This paper finally does a comparison between the economic load dispatch solution with analytical method and using a fuzzy controller. In Fig 8, the cost comparison curve is shown. By adjusting the MFs of the input and output quantities it was possible to bring down the fuel cost significantly but doing it leads to mismatch of the power equation. Hereby we can conclude that the analytical solution is showing better result in comparison to fuzzy method.
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