6. Numerical tests

In order to validate the proposed hybrid algorithm, the CEED problem was solved for IEEE 30 bus system and the results are presented in this section. The fuel cost coefficients with valvepoint loading, emission coefficients, and generator limits are adapted from [26] and is given in Table 1. The transmission loss B-matrix coefficients are obtained by running a load flow program and is in [26] is adapted here and given in Table 2. The total power demand in the system is 2:834 p:u: to the base of 100 MVA. Program in MATLAB was developed for the Hybrid Algorithm to perform CEED and executed on 1:60 GHz, Intel T2050 processor, 1:5 GB RAM HP Pavilion Laptop with WINDOWS 7 operating system. Various test cases are considered to compute the Pareto front of the multi-objective CEED problem. The Pareto-optimal front is obtained using the NSGA II algorithm and also using the MOPSO algorithm given in Section 4. The Pareto front obtained from the hybrid approach given in Section 5 is then compared with the Pareto front obtained using NSGAII and MOPSO algorithm.

In case 1 the fuel cost function is modeled as a quadratic function with sine term to incorporate the valve-point effect. The transmission losses are also considered in this case. The Pareto front obtained using NSGA II, MOPSO, and Hybrid NSGAII-MOPSO is shown in Figures 1, 2 and 3 respectively. In all these figures there is a discontinuity in the Pareto front due to modeling of the valve point loading effect of generators.

The parameter settings for NSGA II are obtained using trial and error is as follows: M ¼ 2; Population Size nPop ¼ 100; Maximum number of iteration MaxIt ¼ 100; Crossover Percentage


Table 1. Fuel costs Coefficients with valve point loading, Emission Coefficients, Generator limits of IEEE 30 bus system.

pCrossover ¼ 0:7; Mutation Percentage pMutation ¼ 0:4; Mutation rate mu ¼ 0:02. The extreme points of the Pareto front and time for execution of NSGAII algorithm are provided

B 0.02180 0.01070 �0.00036 �0.00110 0.00055 0.00330

B0 1.0731e�05 0.0017704 �0.0040645 0.0038453 0.0013832 0.0055503

0.01070 0.01704 �0.00010 �0.00179 0.00026 0.00280 �0.00040 �0.00020 0.02459 �0.01328 �0.01180 �0.00790 �0.00110 �0.00179 �0.01328 0.02650 0.00980 0.00450 0.00055 0.00026 �0.01180 0.00980 0.02160 �0.00010 0.00330 0.00280 �0.00792 0.00450 �0.00012 0.02978

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in Table 3.

B00 0.0014

Table 2. B�Loss Coefficients for IEEE 30 bus test system.

Figure 1. Pareto-optimal curve for IEEE 30 bus system obtained using NSGA II.

Figure 2. Pareto-optimal curve for IEEE 30 bus system obtained using MOPSO.

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Table 2. B�Loss Coefficients for IEEE 30 bus test system.

• Create a new set of particles half the size nPop and fill it with the non-dominated

• Combine the populations of NSGA II and the new set of particles of the MOPSO

In order to validate the proposed hybrid algorithm, the CEED problem was solved for IEEE 30 bus system and the results are presented in this section. The fuel cost coefficients with valvepoint loading, emission coefficients, and generator limits are adapted from [26] and is given in Table 1. The transmission loss B-matrix coefficients are obtained by running a load flow program and is in [26] is adapted here and given in Table 2. The total power demand in the system is 2:834 p:u: to the base of 100 MVA. Program in MATLAB was developed for the Hybrid Algorithm to perform CEED and executed on 1:60 GHz, Intel T2050 processor, 1:5 GB RAM HP Pavilion Laptop with WINDOWS 7 operating system. Various test cases are considered to compute the Pareto front of the multi-objective CEED problem. The Pareto-optimal front is obtained using the NSGA II algorithm and also using the MOPSO algorithm given in Section 4. The Pareto front obtained from the hybrid approach given in Section 5 is then

In case 1 the fuel cost function is modeled as a quadratic function with sine term to incorporate the valve-point effect. The transmission losses are also considered in this case. The Pareto front obtained using NSGA II, MOPSO, and Hybrid NSGAII-MOPSO is shown in Figures 1, 2 and 3 respectively. In all these figures there is a discontinuity in the Pareto front due to modeling of

The parameter settings for NSGA II are obtained using trial and error is as follows: M ¼ 2; Population Size nPop ¼ 100; Maximum number of iteration MaxIt ¼ 100; Crossover Percentage

1 0.05 0.5 10 200 100 15 6.283 4.091 �5.554 6.490 2e�<sup>4</sup> 2.857 2 0.05 0.60 10 150 120 10 8.976 2.543 �6.047 5.638 5e�<sup>4</sup> 3.333 3 0.05 1.00 20 180 40 10 14.784 4.258 �5.094 4.586 1e�<sup>6</sup> 8.000 4 0.05 1.20 10 100 60 5 20.944 5.326 �3.550 3.380 2e�<sup>3</sup> 2.000 5 0.05 1.00 20 180 40 5 25.133 4.258 �5.094 4.586 1e�<sup>6</sup> 8.000 6 0.05 0.60 10 150 100 5 18.480 6.131 �5.555 5.151 1e�<sup>5</sup> 6.667

Table 1. Fuel costs Coefficients with valve point loading, Emission Coefficients, Generator limits of IEEE 30 bus system.

<sup>i</sup> ai bi ci ei fi α<sup>i</sup> β<sup>i</sup> γ<sup>i</sup> η<sup>i</sup> δ<sup>i</sup>

Unit i Generation Limits Fuel Cost Coefficients with valve point loading Emission Coefficients

compared with the Pareto front obtained using NSGAII and MOPSO algorithm.

solutions in the repository followed by the pBest

• Increment generation count

94 Particle Swarm Optimization with Applications

the valve point loading effect of generators.

Pɡmin

<sup>i</sup> <sup>P</sup>ɡmax

• End for

6. Numerical tests

Figure 1. Pareto-optimal curve for IEEE 30 bus system obtained using NSGA II.

Figure 2. Pareto-optimal curve for IEEE 30 bus system obtained using MOPSO.

pCrossover ¼ 0:7; Mutation Percentage pMutation ¼ 0:4; Mutation rate mu ¼ 0:02. The extreme points of the Pareto front and time for execution of NSGAII algorithm are provided in Table 3.

Figure 3. Pareto-optimal curve for IEEE 30 bus system obtained using Hybrid NSGAII and MOPSO Algorithm.

The parameter settings for MOPSO is obtained using trial and error is as follows: M ¼ 2; Maximum number of iteration MaxIt ¼ 500; Population Size nPop ¼ 250; Repository size nRep ¼ 100; Inertia Weight w ¼ 0:5; Inertia Weight damping rate wdamp ¼ 0:99; Personal learning coefficient c1 ¼ 1; Global learning coefficient c2 ¼ 2; Number of grids per dimension nGrid ¼ 10; Inflation Rate alpha ¼ 0:1, leader selection pressure beta ¼ 2, Deletion selection pressure gamma ¼ 2; Mutation rate mu ¼ 0:1. The extreme points of the Pareto front and time for execution of MOPSO algorithm are provided in Table 3. We can observe from Figure 2 and Table 3 that there are difficulties in MOPSO algorithm in obtaining well spread Pareto front and also very slow convergence to the Pareto front when compared to NSGA II. This can be improved if the proposed hybrid approach is used to solve the CEED problem.

algorithm. Even though the time of execution of the Hybrid algorithm is slower than NSGA II it is able to find well spread Pareto front compared to NSGA II. The hybrid algorithm is far superior to MOPSO in terms of converge speed and also in finding well spread Pareto-optimal

Figure 4. Pareto-optimal curve for IEEE 30 bus system without valve point effect obtained using hybrid NSGAII and

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In case II the valve point effect is neglected from the fuel cost curve and is solved using the proposed hybrid approach using the same parameters. The Pareto front obtained is shown in Figure 4 and is a continuous curve when compared to the Pareto front shown in Figure 3. In Figure 3 the Pareto front is discontinuous due to the effect of the Valve point loading in the cost curve. Both these case studies indicate that the hybrid approach is effective to solve the CEED

In this chapter, a hybrid multi-objective optimization algorithm based on NSGA II and MOPSO have been proposed to solve the highly nonlinear, highly constrained combined economic emission dispatch problem. At any stage of the algorithm, only feasible solution is created because of the incorporation of the proposed constraint handling mechanism. During every iteration of the hybrid algorithm new population is created and NSGA II is applied on best performing individuals whereas MOPSO is applied on the lower ranked individuals to strengthen the exploration and exploitation capability of the algorithm. This hybrid approach is tested on an IEEE 30 bus system. The results obtained shows that the hybrid approach is efficient for solving CEED problem and is also able to quickly converge to a better Paretooptimal front when compared to MOPSO algorithm. The result obtained by the hybrid approach also demonstrates it is able to yield a wide spread of solutions and convergence to

front.

MOPSO algorithm.

problem.

7. Conclusion

true Pareto-optimal fronts.

The Parameter setting for the hybrid algorithm is same as those given above expect for the settings provided here Population Size nPop ¼ 200; Maximum number of iteration MaxIt ¼ 50; Repository size nRep ¼ 20. The extreme points of the Pareto front and time for execution of the proposed NSGAII-MOPSO hybrid algorithm are provided in Table 3. From Table 3 it is clear that the extreme points found by the hybrid algorithm are better than NSGA II and MOPSO


Table 3. Comparison of extreme points (shown in bold) and time taken for convergence using NSGAII, MOPSO and Hybrid NSGA II-MOPSO for IEEE30 bus system with valve point loading.

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Figure 4. Pareto-optimal curve for IEEE 30 bus system without valve point effect obtained using hybrid NSGAII and MOPSO algorithm.

algorithm. Even though the time of execution of the Hybrid algorithm is slower than NSGA II it is able to find well spread Pareto front compared to NSGA II. The hybrid algorithm is far superior to MOPSO in terms of converge speed and also in finding well spread Pareto-optimal front.

In case II the valve point effect is neglected from the fuel cost curve and is solved using the proposed hybrid approach using the same parameters. The Pareto front obtained is shown in Figure 4 and is a continuous curve when compared to the Pareto front shown in Figure 3. In Figure 3 the Pareto front is discontinuous due to the effect of the Valve point loading in the cost curve. Both these case studies indicate that the hybrid approach is effective to solve the CEED problem.

## 7. Conclusion

The parameter settings for MOPSO is obtained using trial and error is as follows: M ¼ 2; Maximum number of iteration MaxIt ¼ 500; Population Size nPop ¼ 250; Repository size nRep ¼ 100; Inertia Weight w ¼ 0:5; Inertia Weight damping rate wdamp ¼ 0:99; Personal learning coefficient c1 ¼ 1; Global learning coefficient c2 ¼ 2; Number of grids per dimension nGrid ¼ 10; Inflation Rate alpha ¼ 0:1, leader selection pressure beta ¼ 2, Deletion selection pressure gamma ¼ 2; Mutation rate mu ¼ 0:1. The extreme points of the Pareto front and time for execution of MOPSO algorithm are provided in Table 3. We can observe from Figure 2 and Table 3 that there are difficulties in MOPSO algorithm in obtaining well spread Pareto front and also very slow convergence to the Pareto front when compared to NSGA II. This can be

Figure 3. Pareto-optimal curve for IEEE 30 bus system obtained using Hybrid NSGAII and MOPSO Algorithm.

The Parameter setting for the hybrid algorithm is same as those given above expect for the settings provided here Population Size nPop ¼ 200; Maximum number of iteration MaxIt ¼ 50; Repository size nRep ¼ 20. The extreme points of the Pareto front and time for execution of the proposed NSGAII-MOPSO hybrid algorithm are provided in Table 3. From Table 3 it is clear that the extreme points found by the hybrid algorithm are better than NSGA II and MOPSO

NSGA II 0.0649 0.3866 0.6851 0.7999 0.5399 0.3886 0.03126 616.426 0.2121 367

MOPSO 0.0626 0.4106 0.6885 0.7994 0.5472 0.3564 0.03090 618.211 0.2125 1507

0.4070 0.4528 0.5416 0.4198 0.5365 0.5087 0.03279 677.941 0.1942

0.4412 0.4574 0.5501 0.3821 0.5523 0.4832 0.03242 678.702 0.1943

0.4109 0.4563 0.5429 0.4002 0.5435 0.5128 0.03279 678.30 0.1942

Table 3. Comparison of extreme points (shown in bold) and time taken for convergence using NSGAII, MOPSO and

0.0500 0.3893 0.6861 0.8001 0.5490 0.3911 0.03178 613.85 0.2127 662

(\$/h)

Emission (Tons/h)

Time Taken (s)

improved if the proposed hybrid approach is used to solve the CEED problem.

Method Pɡ<sup>1</sup> Pɡ<sup>2</sup> Pɡ<sup>3</sup> Pɡ<sup>4</sup> Pɡ<sup>5</sup> Pɡ<sup>6</sup> Pl Fuel Cost

Hybrid NSGA II-MOPSO for IEEE30 bus system with valve point loading.

Hybrid NSGAII-MOPSO

96 Particle Swarm Optimization with Applications

In this chapter, a hybrid multi-objective optimization algorithm based on NSGA II and MOPSO have been proposed to solve the highly nonlinear, highly constrained combined economic emission dispatch problem. At any stage of the algorithm, only feasible solution is created because of the incorporation of the proposed constraint handling mechanism. During every iteration of the hybrid algorithm new population is created and NSGA II is applied on best performing individuals whereas MOPSO is applied on the lower ranked individuals to strengthen the exploration and exploitation capability of the algorithm. This hybrid approach is tested on an IEEE 30 bus system. The results obtained shows that the hybrid approach is efficient for solving CEED problem and is also able to quickly converge to a better Paretooptimal front when compared to MOPSO algorithm. The result obtained by the hybrid approach also demonstrates it is able to yield a wide spread of solutions and convergence to true Pareto-optimal fronts.
