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Economic load dispatch (ELD) problems are traditionally solved by convex optimization techniques. However, these techniques are no longer effective if the ELD problem has a nonconvex cost function. This paper aims to find a suitable metaheuristic method to solve the ELD problem with nonconvex cost function. The ELD of generators in a power system with valve point loading effect was taken as the research problem. Then, several metaheuristic optimization techniques were compared in their abilities to find the global optimal solution, namely, the lambda iteration method, the teaching and learning based optimization (TLBO) and the oppositional teaching and learning based optimization (OTLBO). The optimization techniques were thoroughly compared through demonstrations on 6, 10, and 14 units test systems. The results show that the OTLBO outperformed the other algorithms in terms of the global optimal solution. Thus, our research confirms the feasibility and effectiveness of the OTLBO for ELD problems with valve point loading effect.
economic load dispatch (ELD), cost function, oppositional teaching and learning based optimization (OTLBO), valve point loading effect
Nowadays, the electric energy market became more and more competitive so that to survive this current situation, optimal power generation is required to minimize the total power generation cost. ELD determines minimum cost operation of network with dispatching the generation sources to meet the load demand. ELD main objective is to minimize the total generation cost and satisfying the several constraints. Nowadays, generator scheduling is a big problem for power engineers. Since from the past few decades, number of techniques are practiced for economic load dispatch problems. The ELD tells that optimal generator scheduling of loads so that supplying power must be equal to power demanding and power losses as a decreasing fuel cost [1]. Actually the power generation cost is very high. In India the major power is generated from thermal power plants where the running cost is too high. So it is necessary to minimize the power generation cost as well as transmission losses for ELD problems [23]. Many researchers implemented to number of algorithms to solution of economic load dispatch problems.
Simulated & evolutionary programming algorithms which are integrated based and developed for solving the problems of ELD [4]. Barisal et al. has presented a novel optimization method which contains bacterial foraging technique used to solve the ELD Problems [5]. Issarachai et al. implemented an effective novel technique which is based on ant colony method for optimizing ELD problems based on nonsmooth cost functions [6]. Lin et al. [7] developed novel quantum genetic algorithm which is used to solving the ELD problems that having wind power. Seeker optimization technique is used for solving ELD problems which attains human capabilities like understanding and searching [8]. Artificial immune technique which is clonal selection based is applicable to solve the ELD Problems with valve loading effects [9]. Devendra Sharma et al. implemented a hybrid PSO which is based on multiagent technique to solve ELD problems [10] ELD problems include transmission loses, cubic fuel and quadratic fuel cost functions are solved by equal embedded algorithm [11]. MohammadiIvatloo et al. [12] have been implemented to solve the dynamic economic load dispatch problems by using optimality condition decomposition technique.
A novel technique and coding is implemented for power system economical load dispatch problems using effortless hybrid method (EHM) [13]. Subrahmanyam et al. [11] implemented a novel technique which is used to power system economical load dispatch problems with cubic fuel cost function and transmission loses through hybrid partcile swarm optimization technique which is multo agent based. This technique resolves the PSO problems which are randomness, variables tuning and unique solution [14]. Both convex and nonconvex economic dispatch problems of thermal plants are solved by aBBOmDE techinc [15]. A novel technic is proposed to solve the economic dispatch problems using reinforcement learning method [16].
Ongsakul et al. [17] proposed a novel technique to solve the nonconvex economic problems using on Hopfield neural networks technique which hybridbased method. Further, augmented Lagrange Hopfield network was introduced to solve ELD problems with prohibited zones. Basically this method is based on quadratic programming and piecewise quadratic cost function [18]. However these methods are suffering from the excessive iterations and resulting in large competitions. Singh et al. have been formulated and modelled both stochastic and deterministic technique which is improved particle swarm optimization have been developed to solve the economical dispatch problems with environmental effect [19]. This paper explores the new metaheuristic algorithm i.e. oppositional teaching and learning based optimization technique to solve the ELD problems with valve point loading effect. Previously many mathematical programming methods are developed for solving ELD problems in order to get convergence solution. Linear programming techniques are effective but it will applicable only for piecewise linear cost functions. So nonlinear programming approaches have to be implement for solution of nonlinear cost functions. NR based methods cannot solve the equality constraints problems [20].
This paper tells the solution of ELD problem with valve point loading effect by OTLBO algorithm with consideration of transmission losses. In this paper, OTLBO algorithm is implemented for different test systems i.e. 6, 10 and 14 unit test system and also compared with TLBO algorithm. Finally, OTLBO algorithm gives high quality solution for global minimization.
Section 2 describing about problem formulation related ELD Problem
Section 3 discusses the simulation results about proposed optimization technique and also compared with existing techniques and Section 4 explains conclusions from the present work gestions for future investigations.
Load dispatch solutions defines reducing the fuel cost, real power balancing and satisfying the demand of active power. The ELD problem is represented by Savsani et al. [21].
$F C\left(P_{i}\right)=\sum_{i=1}^{N} F_{i}\left(P_{i}\right)$ (1)
Here FC(P_{i}) = overall fuel cost,
$P_{i}$= Power generation of i^{th} thermal generating unit
The fuel cost is quadratic function so it is,
$F_{i}\left(P_{i}\right)=a_{i} P_{g i}^{2}+b_{i} P_{g i}+c_{i}$ (2)
Subjected to
$\sum_{i=1}^{n} P_{i}=P_{D}+P_{L}$ (3)
$P_{i, \min } \leq P_{i} \leq P_{i, \max }$ (4)
2.1 Economic dispatch problem with valvepoint loading effect
Here valve point effect means sum of quadratic function function plus sinusoidal cost function which is represented by Pal et al. [22].
$F_{i}\left(P_{i}\right)=a_{i}+b_{i} P_{i}+c_{i} P_{i}^{2}+e_{i} * \sin \left(f_{i} *\left(P_{i}^{\text {pin }}P_{i}\right)\right)$ (5)
Here e_{i} and f_{i} are generating units reflecting coefficients.
The line losses are represented by
$P_{L}=\sum_{i=1}^{n} \sum_{j=1}^{n} P_{i} B_{i j} P_{j}+\sum_{i=1}^{n} P_{i} B_{0 i}+B_{00}$ (6)
Here B_{ij}, B_{0i} and B_{00} are coefficients of line loss.
The OTLBO algorithm effectiveness and feasibility is tested on standard test systems like 6, 10, 14 and results are also compared with TLBO algorithm as well as Lambda iteration method.
3.1 The proposed algorithm is implemented as per the flow chart
Step 1: Define the system data includes generators fuel cost coefficients, generation limit and demand power.
Step 2: now teacher phase starts and generators mean value will be determined. Obtain the all population size cost value.
Step 3: select the fittest population size and teacher is assigned based on minimum cost.
Step 4: now learner phase starts and improvement of generation due to interaction with different learners.
Step 5: Stop the iteration process if termination criteria satisfies. The number of iterations are represented in this paper is termination criteria. Finally, the global best fitness and corresponding generation is obtained.
Figure 1. Input and output curve with and without valvepoint loading effect a, b, c, d and e are valve points
Figure 1 shows the operating cost characteristics of thermal station generators with and without valvepoints loading Effect.
Figure 2 illustrates the step by step procedure of TLBO algorithm to optimize the ELD problem in the power system network.
Figure 2. TLBO algorithm flow chart
3.2 Test system1: Six unit test system
This case, a nonconvex cost function based 6 thermal units are considered. The proposed method effectiveness is tested on two different load demands 800 and 1263 MW that can be meet by 6 thermal units. The test system data taken from Ref. [23]. In this case population size is assumed as 60. The TLBO & OTLBO load dispatch results are formulated in Table 1. In this case, 25 independent trails have been made with 200 iterations per trail. Based on the performance, three different methods results are compared shown in below Table 1 & 2.
From the Table 1, at load demand of 800MW, the obtained minimum cost by Lambda iteration is 9528.7222\$/h with the power loss of 5.9642MW. The obtained minimum cost by OTLBO method is 9528.7969\$/h with the power loss of 5.9597 MW. The cost obtained by TLBO is 9528.8844\$/h with power loss of 6.0179 MW. From the records, its clearly shows that the obtained minimum cost by all the methods is almost same as the global solution at the load demand of 800 MW.From Table 2, now the power demand of 1263MW, the obtained minimum cost by Lambda iteration method and OTLBO is 15449.8995\$/h with the power loss of 12.9582MW. The minimum cost obtained by TLBO is 15450.6753\$/h with the power loss of 12.8536MW. The cost obtained by Lambda iteration method and OTLBO is same as the global solution.
Figure 3 shows the comparison of convergence characteristics at different populations for different methods. As shown fig x axis represents iterations and y axis represents minimum cost in \$/hr.
3.3 Test system 2: 10unit system
This case, a nonconvex cost function based 10 thermal units are considered. The performance of the proposed methods was demonstrated at two different load demands and that load demands meet by ten thermal units are 1500 and 2000MW. The test data taken from [24]. Here 100 population size is taken. The dispatch results of 10unit system using the proposed methods are given in Table 2. For this test system, trails of 25 independent are made with 300 iterations/trail. Based on data obtained, the comparisons of six thermal units test by different methods are presented in Table 3 & 4.
Figure 3. Comparison of convergence characteristics for different populations
From the Table 3, at load demand of 1500MW the obtained minimum cost by Lambda iteration technique and TLBO is 81130.0325\$/h with the power loss of 49.0223MW. The obtained minimum cost by OTLBO technique is 81129.7603\$/h with the power loss of 49.007MW. From the above records it says clearly that the obtained minimum cost by the OTLBO is the global solution at the load demand of 1500MW.
Table 1. Comparisonal results for 6unit system with load demand of 800 MW
Unit 
P_{D}=800 MW 

Lambda 
TLBO 
OTLBO 

1 
342.2421 
343.4325 
339.6431 
2 
95.4819 
96.5919 
96.5813 
3 
181.9937 
183.1756 
183.2407 
4 
53.6758 
50 
53.9589 
5 
82.5707 
82.8179 
82.5354 
6 
50.0000 
50 
50 
Generation cost in \$/hr 
9528.7222 
9528.8844 
9528.7969 
Power loss in MW 
5.9642 
6.0179 
5.9597 
Table 2. Comparisonal results for 6unit system with load demand of 1263 MW
Unit 
P_{D}=1263 MW 

Lambda 
TLBO 
OTLBO 

1 
447.5038 
444.4068 
447.5038 
2 
173.3182 
170.8177 
173.3182 
3 
263.4628 
263.9355 
263.4628 
4 
139.0653 
146.5230 
139.0652 
5 
165.4734 
166.4267 
165.4733 
6 
87.1347 
83.7436 
87.1347 
Generation cost in \$/hr 
15449.8995 
15450.6753 
15449.8995 
Power loss in MW 
12.9582 
12.8536 
12.9582 
Table 3. Comparisonal results for 10unit system with demand of 1500 MW
Unit 
P_{D}=1500MW  
Lambda 
TLBO 
OTLBO 

1 
43.5706 
43.5706 
45.6086 
2 
60.8157 
60.8157 
61.7683 
3 
72.1301 
72.1301 
67.6629 
4 
60.3987 
60.3987 
55.5074 
5 
51.3367 
51.3367 
51.4848 
6 
71.3367 
71.3367 
71.4848 
7 
207.1676 
207.1676 
209.5246 
8 
222.2243 
222.2243 
232.5880 
9 
372.1789 
372.1789 
375.2049 
10 
387.8631 
387.8631 
378.1727 
Generation cost in \$/hr 
81130.0325 
81130.0325 
81129.7603 
Power loss in MW 
49.0223 
49.0223 
49.0070 
From Table 4, now power demand of 2000MW the obtained minimum cost by Lambda iteration method is 111261.5057\$/h with the power loss of 87.0403MW. The obtained minimum cost by OTLBO method is 11261.5051\$/h with the power loss of 87.0403MW. The TLBO obtained cost is 111289.9482\$/h with power loss of 87.1252. Therefore, the cost obtained by Lambda iteration method and OTLBO is almost same but the cost obtained by OTLBO method is global minimum.
Figure 4 shows the graphical representation of comparison convergence characteristics of obtained minimum cost for 20runs at load demand 2000 MW. As shown in fig the cost obtained by Lambda iteration method is constant for all runs while the other methods are varying.
Table 4. Comparisonal results for 10unit system with demand of 1500 MW
Unit 
P_{D}=2000 MW 

Lambda 
TLBO 
OTLBO 

1 
55.0000 
55.0000 
55.0000 
2 
80.0000 
80.0000 
80.0000 
3 
107.0165 
120.0000 
107.0151 
4 
99.9004 
95.5547 
99.9007 
5 
81.9005 
77.8408 
81.9024 
6 
83.2229 
78.7297 
83.2221 
7 
300.0000 
300.0000 
300.0000 
8 
340.0000 
340.0000 
340.0000 
9 
470.0000 
470.0000 
470.0000 
10 
470.0000 
470.0000 
470.0000 
Generation cost in \$/hr 
111261.5057 
111289.9482 
111261.5051 
Power loss in MW 
87.0403 
87.1252 
87.0403 
Figure 4. Comparison of convergence characteristics of obtained minimum cost for 20runs
3.4 Test system 3: 14unit system
This case, a nonconvex cost function based 14 thermal units are considered. The performance of the proposed methods is demonstrated at two different load demands and that load demands meet by 14 thermal units are 1500 and 2000MW. The data taken from Ref. [25]. Here population is 140. The dispatch results of 14unit system using the proposed methods are given in Tables 5 & 6. For this test system, 500 iterations per trail are made with 25 independent trails. From the data, six thermal units’ comparisons shown by different methods are presented in Tables 5 & 6.
From the Table 5, power demand of 1500MW, obtained minimum cost by Lambda iteration method is 6612.5868 \$/h with the power loss of 17.9213MW. The obtained minimum cost by OTLBO technique is 6612.5089 \$/h with the power loss of 18.1087 MW. TLBO produced the cost of 6612.5120 \$/h with power loss of 18.1655 MW. It has been showed that the minimum cost obtained by all the methods is almost same but the cost obtained by OTLBO is the global solution at the load demand of 1500MW.
From Table 6, now at the power demand of 2000MW, the obtained cost from all the methods is almost same but the cost obtained by OTLBO method is global minimum and it is 8895.4566\$/h with power loss of 30.7713 MW.
Table 5. Comparisonal results for 14unit system with demand of 1500 MW
Unit 
P_{D}=1500 MW 

Lambda 
TLBO 
OTLBO 

1 
221.3101 
220.1858 
218.5729 
2 
189.0354 
192.2743 
190.7673 
3 
50.5688 
49.1485 
53.1257 
4 
88.2294 
86.0241 
88.1582 
5 
150.0000 
150.0258 
150.0026 
6 
135.0000 
135.0258 
135.0026 
7 
135.0000 
135.0258 
135.0026 
8 
60.0000 
60.0258 
60.0026 
9 
139.6414 
139.3569 
136.2976 
10 
127.1018 
130.8644 
132.3087 
11 
79.9875 
80.0000 
80.0000 
12 
79.9875 
80.0000 
80.0000 
13 
47.0593 
45.1827 
43.8651 
14 
15.0000 
15.0258 
15.0026 
Generation cost in \$/hr 
6612.5868 
6612.5120 
6612.5089 
Power loss in MW 
17.9213 
18.1655 
18.1087 
Table 6. Comparisonal results for 14unit system with demand of 2000 MW
Unit 
P_{D}=2000 MW 

Lambda 
TLBO 
OTLBO 

1 
310.6826 
308.1352 
312.1435 
2 
269.8385 
276.6890 
271.6552 
3 
120.5517 
117.5742 
116.8470 
4 
129.9988 
130.0000 
130.0000 
5 
192.9272 
192.3146 
193.8548 
6 
163.3757 
162.2258 
165.2621 
7 
136.9125 
136.0410 
136.1677 
8 
84.6855 
86.0736 
82.8410 
9 
162.0000 
162.0000 
162.0000 
10 
159.9811 
160.0000 
160.0000 
11 
80.0000 
80.0000 
80.0000 
12 
80.0000 
80.0000 
80.0000 
13 
85.0000 
85.0000 
85.0000 
14 
55.0000 
55.0000 
55.0000 
Generation cost in \$/hr 
8895.6328 
8895.5806 
8895.4566 
Power loss in MW 
30.9535 
31.0534 
30.7713 
Figure 5. Distribution of generation conceded by ten generators at P_{D}=1500 MW
Figure 4 shows how the generation shared by fourteen generators with respect to their minimum and maximum limits which means it satisfies the inequality constraint. From the equality constraint, the fourteen generators generation should meet to given load demand.
In this paper, standard ELD problem can be solved in different cases with different methods. In first case ELD problem is represented with nonconvex costfunction, already present there in network. The algorithms TLBO & OTLBO are successfully used to minimize the ELD problem considering 6, 10 and 14unit test systems and also distinguished with lambda technique to test the performance of the proposed algorithm. The proposed algorithm OTLBO found better solution for all test systems than TLBO. This investigation results certainly says that the proposed method can be utilized as effective optimization providing better satisfactory solutions for ELD problems. The paper established algorithms for the ELD problem to have the optimal solution for valve point loading effect only. However i strongly recommend that, in few cases there is still a need to investigate more avenues such as prohibited operating zones, ramp rate limits and multiple fuel selections for each unit. Here only thermal generating units have been considered. The ELD of hydro units can be applied by engaging these novel techniques. I also recommend that the new techniques have been used for combined hydrothermal economic load dispatch for future scope.
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