Oppositional Teaching and Learning Based Optimization of Economical Load Dispatch Problem with Valve Point Loading Effect

Oppositional Teaching and Learning Based Optimization of Economical Load Dispatch Problem with Valve Point Loading Effect

Dsnm RaoChunchu Pushpa Latha Narukullapati Bharath Kumar Perumallu Mahalingam Venkatesh 

Department of Electrical & Electronics Engineering, Gokaraju Rangaraju Institute of Engineering and Technology, Hyderabad 522213, Telangana, India

Department of Electrical and Electronics Engineering, G V R & S College of Engineering & Technology, Guntur 522013, Andhra Pradesh, India

Department of Electrical & Electronics Engineering, VFSTR University, Vadlamudi 522213, A.P, India

Corresponding Author Email: 
2015rsee003@nitjsr.ac.in
Page: 
535-540
|
DOI: 
https://doi.org/10.18280/jesa.520514
Received: 
11 June 2019
|
Accepted: 
13 August 2019
|
Published: 
13 November 2019
| Citation

OPEN ACCESS

Abstract: 

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 non-convex cost function. This paper aims to find a suitable meta-heuristic method to solve the ELD problem with non-convex cost function. The ELD of generators in a power system with valve point loading effect was taken as the research problem. Then, several meta-heuristic 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.

Keywords: 

economic load dispatch (ELD), cost function, oppositional teaching and learning based optimization (OTLBO), valve point loading effect

1. Introduction

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 [2-3]. 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 non-smooth 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 multi-agent 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]. Mohammadi-Ivatloo 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 hybrid-based 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 meta-heuristic 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 non-linear 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.

2. Problem Formulation

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(Pi) = overall fuel cost,

$P_{i}$= Power generation of ith 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 valve-point 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 ei and fi 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 Bij, B0i and B00 are coefficients of line loss.

3. Simulation Results & Discussion

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 valve-point 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 valve-points 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 non-convex 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: 10-unit system

This case, a non-convex 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 10-unit 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 6-unit system with load demand of 800 MW

Unit

PD=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 6-unit system with load demand of 1263 MW

Unit

PD=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 10-unit system with demand of 1500 MW

Unit

PD=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 10-unit system with demand of 1500 MW

Unit

PD=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: 14-unit system

This case, a non-convex 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 14-unit 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 14-unit system with demand of 1500 MW

Unit

PD=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 14-unit system with demand of 2000 MW

Unit

PD=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 PD=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.

4. Conclusion

In this paper, standard ELD problem can be solved in different cases with different methods. In first case ELD problem is represented with non-convex cost-function, already present there in network. The algorithms TLBO & OTLBO are successfully used to minimize the ELD problem considering 6, 10 and 14-unit 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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