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This paper discusses about ELD Problem is modeled by nonconvex functions. These are problem are not solvable using a convex optimization technique. So there is a need for using a heuristic method. Among such methods Teaching and Learning Based Optimization (TLBO) is a newly known algorithm and showed promising results. This paper utilized this algorithm to provide load dispatch solutions. Comparisons of this solution with other standard algorithms like Particle Swarm Optimization (PSO), Differential Evolution (DE) and Harmony Search Algorithm (HSA). This projected algorithm is implemented to resolve the ELD problem for 6 unit and 10 unit test systems along with the other algorithms. This comparison investigation explored various merits of TLBO with respect to PSO, DE, and HSA in the field economic load dispatch.
valve point loading effects, nonconvex, T & L based optimization, PSO, DE, HSA, economic dispatch
As a Power Engineer scheduling the generators is very big Problem. Since from the past so many techniques are in practice for the economic load dispatch. Economic load dispatch means optimal allocation of loads to the generators so as to maintain power supply must be equal to load demand also to decrease the losses and fuel cost (Wood and Wollenberg, 1996). We are all know that power generation is highly costlier. In countries like India the major power generation is form thermal power plants only where the running cost is very high. The one of the best way to minimize the cost and losses of generating station is to Economic dispatch of loads (Amjady and NasiriRad, 2010; Pothiya et al., 2011; Walters and Sheble, 1993). Researchers developed lot of methods for Economic load dispatch. In this work concentrates on an innovative optimization algorithm that is teaching and learning based optimization.
Electrical power plays vital role for any county development. For achieving proper load demand we should have the optimal power flow generation to reduce the cost of production and this can be achieved by economic load dispatch with proper integration of sources to the load centres. The principal goal of Economic Load Dispatch (ELD) is to build effective power flow path while compromising all constraints. The cost function of every alternator can be characterized with quadratic function and it can solve by minimization methods like Lambda iteration and gradient based methods in convention ELD problem (Mahor et al., 2009; Elaiw and Xia, 2010; Chakrabory et al., 2011).
Anciently we developed many methods to clear up the ELD problem like mathematical programming methods and these are more delicate for start and occasionally converge to local optimum solution or diverge altogether. Linear programming approaches are quick and effective however main bad thing is correlated with the piecewise linear cost. Nonlinear programming approaches have a struggle of convergence and algorithmic trouble. Newton based approaches cannot handle many number of equality constraints (Sharifzadeh and Amijady, 2010; Wang, 2013).
This paper explains TLBO algorithm to resolve ELD problem with valve point loading effect of thermal plants by taking transmission losses in to account. We proposed the effectiveness of T&L based Optimization on 6 unit test system and compared with PSO, DE, HSA. Finally T & L based optimization technique gives the high quality solution.
Economic load dispatch means minimizing the fuel cost, balanced Real power, and satisfying real power demand. The ELD problem is shown below (Thanushkodi and Selvakumar, 2007).
$FC({{P}_{i}})=\sum\limits_{i=1}^{N}{{{F}_{i}}}({{P}_{i}})$ (1)
Here, FC(P_{i}) = overall fuel cost,
N = Total number of thermal generating unit,
P_{i} = Power generation of
thermal generating unitThe fuel cost is quadratic function so it is,
${{F}_{i}}({{P}_{i}})={{a}_{i}}P_{gi}^{2}+{{b}_{i}}{{P}_{gi}}+{{c}_{i}}$ (2)
Subjected to $\sum\limits_{i=1}^{n}{{{P}_{i}}={{P}_{D}}+{{P}_{L}}}$ (3)
${{P}_{i,\min }}\le {{P}_{i}}\le {{P}_{i,\max }}$ (4)
Here a_{i},b_{i},c_{i} are fuel cost coefficients of the i^{th} thermal generating unit,
P_{i} = Total true power generation of i^{th} unit
P_{D} = overall load demand,
P_{L} = overall transmission line loss,
P_{i,min} = The minimum generation limit of unit i and
P_{i,max} = The maximum generation limits of unit i.
2.1. Economic dispatch problem with valvepoint loading effect
Here the combination of quadratic and sinusoidal functions of fuel cost to represent the valvepoint loading effects. It follows as (Noman and Iba, 2008; Coelho and Mariani, 2009; Zou et al., 2016; Rao et al., 2011)
${{F}_{i}}({{P}_{i}})={{a}_{i}}+{{b}_{i}}{{P}_{i}}+{{c}_{i}}P_{i}^{2}+\left {{e}_{i}}*\sin ({{f}_{i}}*(P_{i}^{\min }{{P}_{i}})) \right$ (5)
Here e_{i} and f_{i} are coefficient of the generating units reflecting valvepoint loading effects.
The transmission line losses are written as
${{P}_{L}}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{{{P}_{i}}{{B}_{ij}}{{P}_{j}}+\sum\limits_{i=1}^{n}{{{P}_{i}}{{B}_{0i}}+{{B}_{00}}}}}$ (6)
Here B_{ij}, B_{0i} and B_{00} are transmission line loss coefficients.
Teaching and Learning (T&L) inspired optimization process proposed by Rao et al. (2011) and Rao and Patel (2013) depends on Teacher and Learner Mechanism. The Teaching and Learning (T&L) based optimization is a metaheuristic population based search algorithm like HSA, Ant Colony Optimization (ACO), PSO and Artificial Bee Colony (ABC). The Teaching and Learning (T&L) based optimization method is a simple mathematical model to resolve different optimization difficulties.
The projected work concentrates on a new optimization algorithm that is teaching and Learning (T&L) based optimization. Incorporated T&L based optimization algorithm is effective remedy for diminishing the flaws in traditional approach like provincial optimal trapping, inadequate effective to identify adjacent risky points and inefficient appliance to analyzing the constraints. According to our T&L based optimization algorithm a learner can gains knowledge in two ways: (i) by teacher and (ii) interacting with the neighbor learners. In this algorithm beginners are called as population. Design variable are called as subjects of the learners. The top beginner is treated as Teacher.
3.1. Teacher phase
Pupil gains information from the instructor ever and instructor should expand the mean outcome of class by his skills. The best learner is that once knowledge is equal to the teachers knowledge means teacher make to learners to reach his knowledge. But practically is not possible because all learners are not cleverer. This follows as (Kyruakides and Ciornei, 2012)
Let M_{i}= Mean
T_{i} = Teacher at any iteration i.
T_{i} Makes the mean M_{i}to move towards its own knowledge level, therefore T_{i }
chosen as M_{new}. Hence the best learner is treated as teacher. The variance of the current mean result of every subject and the matching result of the teacher for every subject is given by,$Difference=r*({{M}_{new}}{{T}_{F}}{{M}_{i}})$ (7)
Where T_{F}= Teaching factor. It is given as follows:
${{T}_{F}}=round[1+rand*(0,1)*(21)]$ (8)
This difference modifies the existing solution according to the following expression
${{X}_{new,i}}={{X}_{old,i}}+difference$ (9)
3.2. Learner phase
The input for the beginner phase is the teacher in beginner phase learner gains knowledge learner gains knowledge by two ways: one is gaining knowledge form teacher and other is by sharing knowledge between learners interaction.
The learner phase is shows as follows. Randomly select two learners and where i≠j
${{X}_{new,i}}={{X}_{old,i}}+r*({{X}_{i}}{{X}_{j}})$ if $f({{X}_{i}})<f({{X}_{j}})$
${{X}_{new,i}}={{X}_{old,i}}+r*({{X}_{j}}{{X}_{i}})$ if $f({{X}_{i}})>f({{X}_{j}})$ _{ }(10)
Figure 1. Flow Chart of T & L based optimization algorithm
There are several algorithms like PSO, HSA, ABC, GA. The proposed the effectiveness of T&L based Optimization on 6 unit test system and compared with PSO, DE, HSA. Finally, T & L based optimization technique gives the high quality solution.
The Proposed T & L based Optimization algorithm was implemented for two cases case: 1 consisting 6Baseload generation units preferring loading valve point loading effect and losses. The T & L based optimization algorithm was written using MATLAB 8.5 (R2018b) running on i5 processor, 2.56GHz, 8GB RAM, PC.
A. Case 1
This case contains 6base load generation units considering loading valve point loading effect and losses. Generating units have to attain the load demand of 1263MW. To calculate the efficiency of the T & L based optimization method, 25 individual trails can made at 60population with 200 iterations.
Table 1. Global generations for 6unit system per trail
Number of units 
Global generations in MW 

PSO 
HSA 
DE 
TLBO 

1 
400.6115 
399.4068 
500 
500 
2 
199.5996 
200 
149.9957 
151.4009 
3 
232.1225 
232.0630 
230.3581 
300 
4 
124.7998 
125.2627 
125.8899 
87.7215 
5 
199.5996 
200 
149.9629 
149.4573 
6 
120 
120 
120 
88.4572 
Min.cost ($/h) 
15616.7991 
15624.4473 
15615.6937 
15611.6988 
Power loss (MW) 
13.7331 
13.5483 
13.2068 
14.0371 
The comparisons of cost and global are tabulated in Table 1 and Table 2. The global generations and the independent trails convergence characteristics are also plotted which are shown in fig. 2 and 3 respectively.
Table 1 clearly shows that for PSO the minimum cost attained was 15616.7991 \$/h, for HSA the minimum cost attained was 15624.4473 \$/h, for DE the minimum cost attained was 15615.6937\$/h, and for TLBO the minimum cost attained was 15611.6988. Hence the above results shows that, the minimum cost is attained for TLBO as compared with the other algorithms. The power loss attained for TLBO was 14.0371MW.
Figure 2. Convergence characteristics of 6 unit system
Table 2. Minimum cost obtained for 25 runs
Number of runs 
Minimum cost in $/h 

PSO 
HSA 
DE 
TLBO 

1 
15616.8546 
15688.4303 
15635.2652 
15681.9111 
2 
15616.8756 
15677.7093 
15660.2286 
15611.6988 
3 
15758.1765 
15750.0689 
15646.7544 
15680.6254 
4 
15782.4748 
15647.0857 
15645.1185 
15621.5284 
5 
15616.8511 
15657.9900 
15631.8830 
15624.2276 
6 
15625.1855 
15726.5923 
15615.6937 
15621.4526 
7 
15738.7735 
15739.6564 
15632.6176 
15659.3512 
8 
15743.2094 
15647.9531 
15636.6707 
15650.3453 
9 
15626.6348 
15655.4437 
15626.5942 
15650.3141 
10 
15665.8478 
15688.3176 
15673.4684 
15621.5109 
11 
15627.0714 
15703.6266 
15641.7270 
15622.5178 
12 
15616.7991 
15759.3145 
15665.2332 
15621.6119 
13 
15691.2273 
15624.4473 
15652.6820 
15622.4532 
14 
15626.6205 
15656.2226 
15665.7099 
15622.1312 
15 
15616.9367 
15695.9180 
15679.2265 
15621.6684 
16 
15623.5040 
15715.6528 
15638.6161 
15621.6008 
17 
15625.1855 
15740.7103 
15648.2682 
15621.5467 
18 
15626.5741 
15688.7322 
15670.0528 
15621.3824 
19 
15626.7418 
15750.1998 
15629.4167 
15620.9401 
20 
15626.7085 
15769.2848 
15643.9360 
15621.6385 
21 
15618.0267 
15725.9458 
15626.4920 
15622.2550 
22 
15647.0017 
15834.2254 
15639.1709 
15622.9964 
23 
15619.6076 
15751.9471 
15635.1169 
15621.7541 
24 
15623.5005 
15744.5482 
15633.0052 
15622.5070 
25 
15624.3020 
15694.8515 
15637.5919 
15621.6983 
Min. cost ($/h) 
15616.7991 
15624.4473 
15615.6937 
15611.6988 
Max. cost ($/h) 
15782.4748 
15834.2254 
15679.2265 
15681.9111 
Avg. cost ($/h) 
15649.2276 
15709.3950 
15644.4216 
15630.0667 
Figure 3. Comparison characteristics of minimum cost Obtainedfor 25 runs
This case consists of ten thermal generation units considering loading valve point loading effect and losses. Generating units have to attain the load demand of 2000 MW. To calculate the efficiency of the T & L based optimization method, 25 individual trails can ready at 100population with 200 iterations per trail.
The comparisons of cost and global are tabulated in Table 3 and Table 4. The global generations and the independent trails convergence characteristics are also plotted which are shown in fig 4 and 5 respectively.
B. Case 2
Table 3. Global generations for 10unit system
Number of units 
Global generation in MW 

PSO 
HSA 
DE 
TLBO 

1 
55 
50.8495 
55 
55 
2 
80 
75.8420 
78.7733 
80 
3 
107.3388 
115.8420 
99.3983 
106.9392 
4 
100.3117 
94.02348 
107.1068 
100.5765 
5 
81.4700 
109.7019 
89.0972 
81.5012 
6 
82.9208 
95.2030 
81.4078 
83.0217 
7 
300 
295.8420 
296.1400 
300 
8 
340 
335.8420 
340 
340 
9 
470 
465.8420 
470 
470 
10 
470 
446.8475 
470 
470 
Min.cost ($/h) 
111497.6596 
111907.4666 
111537.6219 
111497.6301 
Power loss (MW) 
87.0414 
85.8360 
86.9237 
87.0387 
Figure 4. Convergence characteristics of 10unit system
Table 3 shows that for PSO the minimum cost attained was 111497.6596\$/h, for HSA the minimum cost attained was 111907.4666\$/h, for DE the minimum cost attained was 111537.6219\$/h, and for TLBO the minimum cost attained was 111497.630. Hence the above results shows that, the minimum cost is attained for TLBO as compared with the other algorithms. The power loss attained for TLBO was 87.0387MW.
Table 4. Minimum cost values for 25 runs
Number of runs 
Minimum cost in $/h 

PSO 
HSA 
DE 
TLBO 

1 
111641.4441 
111959.2697 
111569.1983 
111500.9854 
2 
111525.8322 
112694.2246 
111673.5325 
111505.7236 
3 
111497.6763 
111947.6861 
111695.2852 
111497.6765 
4 
111521.5108 
112047.7053 
111567.3306 
111521.7364 
5 
111525.8275 
112302.8949 
111742.5223 
111525.7565 
6 
111525.6877 
112206.2944 
111743.0718 
111521.5768 
7 
111525.7571 
112052.4801 
111670.3818 
111502.6754 
8 
111525.7976 
112071.9085 
111705.6591 
111505.8768 
9 
111525.8834 
111947.8623 
111751.1809 
111497.6301 
10 
111497.7631 
111987.3196 
111648.195 
111497.6764 
11 
111497.6695 
111919.8793 
111645.2498 
111497.6765 
12 
111497.7148 
112337.6419 
111601.2568 
111497.6987 
13 
111497.6784 
112250.1165 
111689.5033 
111497.6877 
14 
111525.7557 
112185.1190 
111663.6215 
111500.6301 
15 
111497.8285 
112235.6711 
111679.4047 
111504.6375 
16 
111497.7403 
112094.2826 
111654.574 
111525.6384 
17 
111525.6996 
112026.1773 
111629.5029 
111518.6311 
18 
111525.7043 
112125.7557 
111537.6219 
111499.6343 
19 
111525.5897 
112010.5037 
111706.3123 
111497.6301 
20 
111525.8344 
112131.3220 
111714.4087 
111497.6301 
21 
111525.7345 
112421.2877 
111551.2658 
111497.6301 
22 
111525.7724 
112461.9869 
111675.4585 
111499.6383 
23 
111497.6596 
112385.1277 
111707.5187 
111499.6376 
24 
111525.71 
112111.6850 
111608.6125 
111497.6301 
25 
111497.7123 
111907.4666 
111652.1783 
111497.6301 
Min cost($/h) 
111497.6596 
111907.4666 
111537.6219 
111497.6301 
Max. cost($/h) 
111641.4441 
112694.2246 
111751.1809 
111525.7565 
Avg.cost($/h) 
111520.1193 
112152.8667 
111659.3138 
111504.2789 
Figure 5. Comparison features of minimum cost obtained for 25 runs
Hence form the above results we can conclude that Incorporated T & L based optimization algorithm is Effective remedy for diminishing the flaws in traditional approach like provincial optimal trapping, inadequate effective to identify adjacent extreme points and inefficient mechanism to analyzing the constraints. The proposed T&L based optimization on 6 unit test system, 10 unit test system compared with PSO, DE, HSA. Finally TL based optimization technique gives the Effective high quality solution for Economic load dispatch problem.
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