Economic dispatch has a significant effect on optimal economical operation in the power systems in industrial revolution 4.0 in terms of considerable savings in revenue. Various non-linearity are added to make the fossil fuel-based power systems more practical. In order to achieve an accurate economical schedule, valve point loading effect, ramp rate constraints, and prohibited operating zones are being considered for realistic scenarios. In this paper, an improved, and modified version of conventional particle swarm optimization (PSO), called Oscillatory PSO (OPSO), is devised to provide a cheaper schedule with optimum cost. The conventional PSO is improved by deriving a mechanism enabling the particle towards the trajectories of oscillatory motion to acquire the entire search space. A set of differential equations is implemented to expose the condition for trajectory motion in oscillation. Using adaptive inertia weights, this OPSO method provides an optimized cost of generation as compared to the conventional particle swarm optimization and other new meta-heuristic approaches.
Management of energy would-be highly effective and efficient by optimizing the generating cost of fossil fuel-based systems. Economic operation of the power system with effective and reliable generation is highly essential for Industry 4.0, as the electricity market is moving towards the deregulated market. The generation cost of thermal power plants mostly relies on fuel cost. Economic load dispatch is a process of economic scheduling of generating power from each generator to meet the demand and attain the optimum fuel cost by considering various constraints [1]. Economic Dispatch is an optimization problem-solving method where the entire requisite generation is being dispersed amongst the operated generating units, by reducing the consumed fuel cost, considering equality, inequality and operational constraints. ELD governs the output power of every generating division for the specific system under a specified load condition by minimizing the fuel cost to meet the load demand. ELD processes such a real-time management of energy in the current power system to control, assign, and distribute the total generation among the accessible units [2]. Generating units used different types of fuel for power generation. During the practical operation, the spinning reserve constraints make a significant impact on financial planning. Considering all the constraints, the economic dispatch problem behaves as a non-convex, complex, and non-smooth optimization problem.
In recent years, the generator processed for the generation of electricity is non-linear as compared to the customary generator. The non-linearity are by the concern of valve point loading, prohibited operating zones, and ramp rate limit. The practical economic dispatch (ED) also satisfies the problems due to non-linearity, non-convexity, and non-smooth operation of the generator. Previously for solving ELD problem, some classical and conventional methods such as lambda iteration [3], quadratic programming [4], gradient programming [5], and non-linear programming [6] others were applied. These classical methods face many challenges during the problem solving of ELD with non-linearity. To overcome the challenges, many meta-heuristic approaches, swarm evolutionary methods, and evolutionary computing methods were applied to the problem. These methods are Particle swarm optimization [7], Genetic Algorithm [8], Differential Evolution [9], Exchange Market Algorithm [10], Social Spider Algorithm [11], Biogeography based optimization [12], Tabu Search Method [13], Particle Diffusion [14], Artificial Bee Colony [15], Grey wolf optimization [16,17] and Spotted Hyena Optimization [18]. Some of the original methods get stock in the local optima and take more time for the searching process; therefore, to improve the quality of the solution, many hybrid techniques were proposed to overcome the difficulties. Recently some hybridized and modified version of existing techniques are applied to ELD problem such as Differential evolution-biogeography based optimization (DE-BBO) [19], Differential Evolution and harmony search (DHS) [20], Hybridization of Genetic Algorithm with Differential evolution (HDEGA) [21], Combination of Simulated Annealing and PSO (SAPSO) [22], Particle Swarm Optimization and Sequential Programming technique (PSO-SQP) [23], Hybrid Chemical Reaction optimization and Differential Evolution Algorithm (CRO-DE) [24], Multi-objective Spotted Hyena Optimizer and Emperor Penguin Optimizer (MOSHEPO) [25], Hybrid Firefly and Genetic algorithm [26], Adaptive real coded genetic algorithm (ARCGA) [27], Improved harmony search (IHS) [28], Modified differential evolution (MDE) [29], Species-based Quantum Particle Swarm Optimization (SQPSO) [30], Modified particle swarm optimization [31], Improved Differential Evolution [32], Modified Artificial Bee Colony algorithm [33], Modified Bacterial Foraging Algorithm (MBFA) [34], Improved Harmony Search with Wavelet Mutation (IHSWM) [35], DHIMAN Algorithm [36] . In addition, authors in [37] propose a Clustering-based Travel Planning System while a network route optimizations scheme is proposed in [38]. According to No Free Lunch theorem (NFL) [39], no optimization technique will be able to claim as the superior optimized solution for the concerned problem. The existence for improvement of cost for economic dispatch problem encourages to further improving the quality of solution regarding optimum cost and convergence property. Each applied technique to get the solution of the economic dispatch problem is having some advantages and disadvantages.
In this paper, an improved version of conventional Particle Swarm Optimization (PSO) is applied to overwhelm certain difficulties. Oscillatory PSO instincts the particle to acquire the total search space for finding optimum cost in ELD problem by balancing exploration and exploitation perfectly. An actual setting of parameter attains the optimum cost. The selections of cognitive and social learning factors are taken by confirming that the divergence does not occur before the optimum cost.
Problem Formulation
The formulated problem of ELD is the economic scheduling of electric power among the committed generating units to satisfy the load demand and satisfying various constraints. The major objective is to optimize the cost of fuel by generator scheduling [15].
Objectives for Economic Load Dispatch Problem
The characteristics of every generator are unique with respect to cost. The steam valve controls the operation of the turbine for the generation of power and is known as the valve point loading effect. This practical approach due to valve point loading characteristics curve of the generator becomes non-convex curve. The cost curve behaves as a piecewise linear increasing quadratic function as shown in Fig. 1. The fuel cost function is dependent on the real power generation from each unit and is shown in Eq. (1) [15].
Cost characteristics of fossil fuel-based generator
Here F_{CC} is the fuel cost of all committed generator. F(P_{genc}) is the cost function of the generator. A_{genc}, B_{genc } and C_{genc }are the cost co-efficients and, E_{genc} and F_{genc} are the co-efficients due to the valve point loading effect. P_{genc} is the scheduled output power. The cost function for multiple fuel options is represented in Eq. (2) [40].
A_{gencj}, B_{gencj}, C_{gencj},E_{gencj} and F_{gencj} are the cost co-efficients for genc number of generating units and ‘j’ type of fuel.
Constraints
The economic scheduling of the generator should have to satisfy the practical operational constraints.
Power Balance Constraint
This is an equality constraint. In a given period, the total scheduled output power of committed generators should satisfy the load estimated following electricity demand and the transmission line losses in the power system [16].
∑genc=1NPgenc−Pde−PLoss=0
Here P_{genc} total scheduled power generation. P_{de} is forecasted load demand by the consumer. P_{Loss} is Transmission line loss. P_{Loss} is expressed in terms of B co-efficient by using Eq. (4) [16].
Real power generates at the output of the generator should be within a prescribed limit higher and basic limit as shown in Eq. (5) [16].
Pgencmin<Pgenc<Pgencmax
P_{gencmain} and P_{gencmax} are the minimum and higher bound for the generation of power.
Generator Ramp Rate Limit
The operational performance for generating units is reserved by ramp rate limits. These limits influencefunctional decisions. The present scheduling may interrupt the upcoming scheduling as a generation grows due to ramp rate bounds [19].
The generator performance is having some discontinuous portions due to some unsought and uncontrollable physical restrictions such asmechanical lossesorfailures. The generator discontinuities are shown in Fig. 2 and Eq. (7) [19].
Generator characteristics with existing prohibited zonesParticle Swarm Optimization
The approached method is a swarm intelligence method based upon the process of collection of food by bird and fish. PSO works in the mechanism of birds to search for food randomly in a specified region. The key approach is to detect food with a reduced time [41]. This approach is based on the procedure to get the food and to observe the bird nearer to the food. The orthodox PSO learned from the condition and handled it to resolve the course to achieve an optimum value. Each bird is an alone solution in the total search space is known as a particle. All the particles are assessed by their corresponding fitness function, which is to be optimized. All the particles in the search space are having their velocities to search for the direction of food.
The initialization of PSO was done by using an arbitrary particle, which is the solution to find the optimal position by the process of updating during generations. During each reiteration course, the entire solutions particles are updated with two optimum values: (1) The finest value among the whole particles obtained by searching the food known as global best and, (2) the finest value monitored by the swarm itself during exploration in repetition process known as personal best. During the process of searching food, the velocity of the bird is to be maintained, by using the following formula by Eq. (8) [42].
In the above equation, V_{k}(i + 1) and V_{k}(i) is the velocity component and rand()×(pebestk−Pk(i)) is particle memory inspiration & rand()×(gebestk−Pk(i)) is swarm inspiration. V_{k}(i) is the velocity of k^{th} particle at iteration (i) must lie in the range of velocity with upper and lower bounds.
Vmin≤Vk(i)≤Vmax
V_{max} and V_{min } are the velocity indicesfor upper and lower boundaries of the particle to move in the search space to locate the food. If V_{max} is extremely high, the particles have a chance to past better solutions. If V_{min} is much small, then particles have a chance not to discover further than local optima. C_{e1} and C_{e2} are two constants to attract each solution towards the best among individual and whole particle locations.
The weight of inertia (w) follows the equation,
we=wemax−{wemax−weminitermax}×iter
where w_{e}, w_{emax}, w_{emin} are the weight of inertia and (iter) is the iteration number.
Oscillatory Particle Swarm Optimizer
In this algorithm, the update equation of the conventional PSO isspecified as a differential equation of second order. The characteristics of convergence are resultant of social and cognitive learning rates. The particle transitional activities dependency on the inertia weight is discovered. Further, the induced oscillation feature and adaption of weight are derived.
Updating PSO as a Differential Equation of Second Order
In the conventional PSO, the velocity and position as per the above Eqs. (8) and (9) is processed. By reducing the iteration count, i + 1 to i the velocity particle is like Vk(i)=Pk(i)−Pk(i−1).
The updated position is represented in the expansion form as in Eq. (12):
Rearranging the above Eq. (13) can be rewritten as:
Pk(i+1)+Po1Pk(i)+Po2Pk(i−1)=Rk
Here the coefficients are Po1=Ce1+Ce2−we−1, Po2=we, Rk=Ce1Pebestk+Ce2gebestk.
Factors of Cognitive and Social Learning
From Eq. (14), the coefficients P_{o1} and P_{o2} determine the particle behavior. Assume the best position of a particle and global as P_{ebestk} and g_{ebestk} respectively remain constant, and both are equal for two successive iterations as shown in Eq. (15). One particle is having the personal best as the global best and the Eq. (15) can be rewritten as Eq. (16) as, Pk∗=Pebestk∗=gebestk∗.
Pk∗+Pk1∗+Pk2∗=Ce1Pebestk∗+Ce2gebestk∗
(1+Po1+Po2)Pk∗=Ce1Pk∗+Ce2Pk∗=(Ce1+Ce2)Pk∗
1+Po1+Po2=Ce1+Ce2
Letting Po1+Po2=0 then, Ce1+Ce2=1, their trajectories satisfy
Here the right-hand side shows the weighted sum of particle best and global best. Consider the complementary equation.
Pk(i+1)+Po1Pk+Po2Pk(i−1)=0,
In the last iteration, the best particle value (Pebestk) and best global value (gebestk) were considered as the optimal solution. The complexity of the applied algorithm was reduced by considering the social and cognitive leaning rates of personal and global best as Ce1+Ce2=1
Weight of Inertia in OPSO
When the cognitive and social learning factor is Ce1+Ce2=1, the coefficients become as per Eq. (20) and the complementary equation is Eq. (21). Here the inertia weight shows the convergence property of the particle trajectories while moving forward the iteration.
Po1=Ce1+Ce2−we−1=−we
Pk(i+2)−wePk(i+1)+wePk(i)=0
For oscillating condition, considering the characteristics Eq. (21) with roots are shown in Eqs. (22) and (23) respectively.
λo2+Po1λo+Po2=0
λo1,2=−Po1±Po12−4Po22
Applying De Moivre's formula for the condition Po12<4Po2, the particle P_{k}(i) will be shown in Eq. (24).
Pk(i)=ri[b1cosiθ+b2siniθ]
Another phase angle φ is presented in Eq. (25).
ϕ=tan−1(b1b2)
cosϕ=b1/b12+b22
sinϕ=b2/b12+b22
The homogeneous equation solution will be shown in Eq. (26).
Pk(i)=rib12+b22[cosiθcosϕ+siniθsinϕ]
=Bricos(iθ−ϕ)
where B=b12+b22 and the P_{k}(i) oscillates for the term cos(iθ−ϕ) in Eq. (26).
From Eq. (26), P_{k}(i) will converge for r < 1 and ‘r’ will be
r=Po124+4Po22−Po124=Po2=we
The oscillatory behavior of particle is governed by the amplitude an phase angle. The frequency of oscillation is determined by angles of the complex roots of the characteristics equation. And by substituting Po1 = −we and Po2 = we, the angle of the root is given byte Eq. (28).
θ1,2=tan−1(±4we−we2we)
Determination of inertia weight can be done by applying normal distribution to the random inertia weights according to Eq. (29).
w˜e=N(we,σ),σ=we
In this method the inertia weight is calculated by using the Eq. (30).
we(i)=wein−weend×iimax,wk=N(wek,wek)
The detail pseudo of the applied algorithm for economic load dispatch is given in Algorithm~1.
Step 1
Initialize the no. of Iterations i_{max}, Population, Particles, Velocity, Start and End inertia weight
Step 2
Convential PSO
For i = i_{max} do()
Find the fitness function (Cost) for each particle (Generator)
Identify the best fitness value (Cost) for each particle (P_{ebest}) and global best (g_{ebest})
Oscillatory PSO
Generate the social learning factor C_{e2}, and Cognitive learning factor C_{e1} = 1 − C_{e2}
Calculate the inertia weight using Eq. (29)
Update the Particle Velocity and Particle Position
endfor
Step 3
Best fitness value find (Optimum Generation Cost) due to the best particle is found
Results and Discussion
The proposed technique is applied to optimize the overall cost of four different test systems within the framework of different linear and non-linear technical constraints and multiple fuel systems. The four test systems considered in this study are: (1) A 6-unit system considering the transmission losses, (2) A 15-unit system considering the transmission losses (3) A large power system with 40 generating units considering valve-point loading and, (4) A ten-unit test system for different types of fuel. The simulation is performed on the MATLAB (version R2016b) platform. A total of 100 runs have been executed for generating an optimum solution to the discussed dispatch problem.
Case 1: Six generating Units test system
This case consists of six generating units to fulfill the load demand of 1263 MW. As transmission losses make a huge impact on the power system, transmission losses, prohibited operating zones, ramp rate limits, and valve point loading effect are considered. The system input data for all the constraints, cost co-efficient, and loss co-efficient are considered from [7]. The scheduled generation among all the six generating units within the capacity constraint with the optimum cost is presented in Tab. 1. The comparison of cost and scheduled generation with other techniques is also presented in Tab. 1. Fig. 4 shows the comparison graph of optimum cost with other techniques to validate the superiority of the applied technique. The optimum cost for this test system is found as 15,440.0982 $/h with a lesser transmission loss of 12.178 MW. The convergence graph is shown in Fig. 3. Prohibited operating zones, valve point loading, and ramp rate constraints are also considered for the complex problem. All the input data and co-efficient are referred to from [7] for this test system.
Comparison table for scheduled generation and optimum cost for case 1 with losses
Method
{ P1 (MW)}
{ P2 (MW)}
{ P3 (MW)}
{ P4 (MW)}
{ P5 (MW)}
{ P6 (MW)}
{ Total generation (MW)}
{ Total loss (MW)}
{ Generation cost ($/h)}
BSA [43]
447.4902
178.3308
263.4559
139.0602
165.4804
87.1409
1275.9583
12.9583
15449.8995
BFO [44]
449.4600
172.8800
263.4100
143.4900
164.9100
81.2520
1275.4020
12.4020
15,443.8497
CRO [45]
447.9314
173.5548
262.9452
138.8521
165.3046
86.8575
1275.4456
12.4456
15,443.080
HCRO-DE [24]
447.4021
173.2407
263.3812
138.9774
165.3897
87.0538
1275.4449
12.4449
15,443.0750
MIQCQP [46]
447.4000
173.2400
263.3800
138.9800
165.3900
87.0500
1275.4400
12.4400
15,443.0700
IABC-LS [47]
451.5204
172.1750
258.4186
140.6441
162.0797
90.3415
1275.1795
12.1795
15,441.1080
DHS [20]
447.5285
173.2791
263.4772
139.0291
165.4864
87.1587
1275.9590
12.9590
15,449.8996
SQ-PSO [30]
446.7273
173.4511
263.5318
138.9152
165.4092
87.2577
1275.2923
12.4422
15,441.0497
NPSO [48]
447.4734
173.1012
262.6804
139.4156
165.3002
87.9761
1275.9500
12.9571
15,450.0000
Ɵ–PSO [49]
445.5434
171.5376
263.0251
138.6269
165.6061
91.1055
1275.4446
12.4459
15,443.2717
MPSO-GA [50]
444.3230
173.1810
265.0000
140.3290
166.1200
86.4210
1275.3770
12.3700
15,442.4640
ICS [51]
447.6162
173.5795
262.7578
139.1206
165.6426
86.6658
1275.3800
12.3924
15,442.2652
APSO [52]
446.6690
173.1560
262.8260
143.4690
163.9140
85.3440
1275.3800
12.4220
15,443.5800
PSO
447.4970
173.3221
263.4745
139.0594
165.4761
87.1280
1276.0100
12.9583
15,450.0000
MPSO
446.4870
168.6610
265.0000
139.4930
164.0040
91.7470
1275.3900
12.3740
15,443.1000
OPSO
451.518
172.175
258.413
140.644
162.078
90.342
1275.17
12.178
15,440.0982
Convergence characteristics for case 1 with 1263 MW load demandComparison graph for six generating units with other applied techniquesComparison table for an optimum cost for 15 generating units with variation
Techniques
{ Best cost ($/h)}
{ Average cost ($/h)}
{ Worst cost ($/h)}
{ Output power (MW)}
{ CPU time (s)}
ICS [51]
32,706.7358
32,714.4669
32,752.5183
2660.734
–
Ɵ-PSO [49]
32,706.5504
32,738.0235
32,707.6065
2660.8213
36.88
MIQCQP [46]
32,704.58
–
–
2660.66
4.65
MPSO-GA [50]
32,702
32,733.29
32,755.19
2660.034
NRTO [53]
32,701.81
32.704.53
32,715.18
2660.42
29.38
RCCRO [54]
32,698.9950
32.698.995
32,698.995
2658.7040
4.0
MBBO [55]
32,692.3972
32,692.3973
32,692.3975
2659.5848
–
OLCSO [56]
32,692.3961
32,692.3981
32,692.4033
2659.5846
–
DEPSO [57]
32,588.81
32,588.99
32,591.49
2657.966
–
λ-Con [58]
32,568.06
–
–
2659.60
–
KGMO [59]
32,548.1736
32,548.2163
32,548.3755
2656.8983
7.24
PSO
32,705.3214
32,812.6654
32,922.3274
2660.479
10.25
MPSO
32,554.365
32,614.9854
32,662.3765
2658.586
7.41
OPSO
32,548.021
32,549.2541
32,549.9874
2656.899
4.12
Case 2: Fifteen generating units test system with transmission losses
Fifteen generating units are used for the generation of demand of 2630 with consideration of transmission losses, in the test generation for optimum cost from the applied technique with comparison to other techniques. Tab. 2 shows the evidence of the superiority of the applied technique for optimum cost with lesser variation during the iteration process as compared to MPSO-GA [50], NRTO [53], MsBBO [55], DEPSO [57], λ-Con [58], ICS [51] techniques. Fig. 5 represents the optimum cost comparison of the applied technique to the other techniques and Fig. 6 represents the convergence characteristics of the applied technique with the conventional PSO. Tab. 3 compares scheduled generation and optimum cost for case 2 with losses.
Comparison graph for fifteen generating units with other techniquesConvergence characteristics of case 2 for fifteen generating unitsComparison table for scheduled generation and optimum cost for case 2 with losses
Method
DEPSO [57]
WCA [60]
DHS [20]
λ-Con [58]
EMA [10]
OLCSO [56]
MPSOGA [50]
P1 (MW)
455.000
455.000
455.0000
455.0000
455.0000
455.0000
455.0000
P2 (MW)
420.000
380.000
420.0000
455.0000
380.0000
380.0000
380.0000
P3 (MW)
130.000
130.000
130.0000
130.0000
130.0000
130.0000
130.0000
P4 (MW)
130.000
130.000
130.0000
130.0000
130.0000
130.0000
130.0000
P5 (MW)
270.000
170.000
270.0000
298.2294
170.000
170.000
169.9600
P6 (MW)
460.000
460.000
460.0000
460.0000
460.0000
460.0000
460.0000
P7 (MW)
430.000
430.000
430.0000
465.0000
430.0000
430.0000
430.0881
P8 (MW)
60.000
71.721
60.0000
60.0000
74.042
69.4738
60.1300
P9 (MW)
25.000
58.941
25.0000
25.0000
58.621
60.1108
72.6064
P10 (MW)
62.966
160.000
62.9762
25.0000
160.000
160.0000
157.0093
P11 (MW)
80.000
80.0000
80.0000
44.9350
80.0000
80.0000
80.0000
P12 (MW)
80.000
80.0000
80.0000
56.4370
80.0000
80.0000
79.2381
P13 (MW)
25.000
25.0000
25.0000
25.0000
25.0000
25.0000
26.0017
P14 (MW)
15.000
15.0000
15.0000
15.000
15.0000
15.0000
15.0000
P15 (MW)
15.000
15.0000
15.0000
15.000
15.0000
15.0000
15.0000
Total generation (MW)
2657.966
2660.66
2657.9762
2659.60
2660.66
2659.5846
2660.034
Total loss (MW)
27.976
30.66
27.9762
29.60
30.66
29.5846
29.4031
Cost ($/h)
32,588.81
32,704.449
32588.9182
32,568.06
32,704.450
32,692.3961
32,702
Method
NRTO [53]
MIQCQP [46]
MsBBO [55]
Ɵ–PSO [49]
PSO
MPSO
OPSO
P1 (MW)
455.0000
455.0000
455.0000
455.0000
455.0000
454.8914
454.8229
P2 (MW)
380.0000
380.0000
380.0000
380.0000
380.0000
454.8914
449.0101
P3 (MW)
129.9999
130.0000
130.0000
130.0000
129.9998
129.9997
129.4101
P4 (MW)
129.9999
130.0000
130.0000
130.0000
129.9998
129.9997
129.9999
P5 (MW)
170.0000
170.000
170.000
170.000
170.000
235.7547
239.7498
P6 (MW)
460.0000
460.0000
460.0000
460.0000
460.0000
459.9632
459.5598
P7 (MW)
430.0000
430.0000
430.0000
430.0000
430.0000
464.9668
464.9799
P8 (MW)
70.2250
72.13
69.4798
71.8045
70.3544
60.3255
61.2211
P9 (MW)
60.1965
58.54
60.1049
60.2379
60.1247
25.3741
25.5999
P10 (MW)
159.9999
160.00
160.0000
158.7524
159.9998
29.3001
28.1127
P11 (MW)
80.0000
80.0000
80.0000
80.0000
80.0000
77.7147
78.7451
P12 (MW)
80.0000
80.0000
80.0000
80.0000
80.0000
80.0201
80.3658
P13 (MW)
25.0000
25.0000
25.0000
25.0078
25.0000
25.3741
25.3214
P14 (MW)
15.0000
15.0000
15.0000
15.0147
15.0000
15.0010
15.0001
P15 (MW)
15.0000
15.0000
15.0000
15.0040
15.0000
15.0090
15.0001
Total generation (MW)
2660.4216
2660.66
2659.5848
2660.8213
2660.479
2658.586
2656.899
Total loss (MW)
30.4216
30.66
29.5848
30.8319
30.479
28.586
26.899
Cost ($/h)
32701.8145
32,704.45
32692.3972
32,706.5504
32,705.3214
32,554.365
32,548.021
Case 3: Test system 3 for forty generating units
This test system is considered for a large power system consisting of 40 generators. In this case, the effect of valve-point loading is considered as the non-linear constraint. The input data is referred from [61] for the co-efficient and various load demands. The scheduled generation among 40 units to meet the total demand of 10500 MW is illustrated in Tab. 4. Tab. 5 shows a comparison with other recent techniques for minimum cost. The deviation of the costs among different optimization techniques along with the proposed technique is presented in Tab. 5. The convergence graphs and cost comparison are shown in Figs.7 and 8 respectively. It is analyzed from this case study that the applied OPSO algorithm provides better performance for minimizing the cost of the power supply, losses, and the convergence time as compared to the existing optimization techniques under the considered operating condition.
Scheduled generation among 40 generators to satisfy the demand for optimum cost
UNIT
{ Output power (MW)}
UNIT
{ Output power (MW)}
UNIT
{ Output power (MW)}
UNIT
{ Output power (MW)}
P1 (MW)
110.7987
P11 (MW)
94.0000
P21 (MW)
523.2803
P31 (MW)
190.0000
P2 (MW)
110.7987
P12 (MW)
94.0000
P22 (MW)
523.2801
P32 (MW)
190.0000
P3 (MW)
97.4121
P13 (MW)
214.7602
P23 (MW)
523.2801
P33 (MW)
190.0000
P4 (MW)
179.7411
P14 (MW)
394.2833
P24 (MW)
523.2803
P34 (MW)
164.7875
P5 (MW)
87.8099
P15 (MW)
394.2833
P25 (MW)
523.2801
P35 (MW)
199.9988
P6 (MW)
140.0000
P16 (MW)
394.2833
P26 (MW)
523.2803
P36 (MW)
194.3198
P7 (MW)
259.6018
P17 (MW)
489.2801
P27 (MW)
10.0000
P37 (MW)
110.0000
P8 (MW)
284.6121
P18 (MW)
489.2801
P28 (MW)
10.0000
P38 (MW)
110.0000
P9 (MW)
284.6008
P19 (MW)
511.2811
P29 (MW)
10.0000
P39 (MW)
110.0000
P10 (MW)
130.0000
P20 (MW)
511.2811
P30 (MW)
87.8184
P40 (MW)
511.2866
Total generation (MW)
10,500.00
Cost ($/h)
1,21,410.3232
Comparison of cost with deviation for 40 generators in test system 3
Techniques
{ Best cost ($/h)}
{ Average cost ($/h)}
{ Worst cost ($/h)}
{ Output power (MW)}
{ CPU time (s)}
AA [62]
121,788.70
–
–
10500
–
CPSO–SQP [61]
121,458.54
122,028.16
–
10500
98.49
THS [63]
121,425.15
121,528.65
–
10500
–
Ɵ–PSO [49]
121,420.90
121,509.84
121,852.42
10500
CRO [45]
121,416.69
121,418.03
121,422.92
10500
8.15
DFA [64]
121,414.64
121,415.78
121,422.12
10500
–
C-GRASP [65]
121,414.621
–
–
10500
–
NTHS [66]
121,412.74
121,549.95
–
10500
–
IABC [47]
121,412.73
–
121,471.61
10499.9999
8.76
DE [67]
121,412.68
121,439.89
121,479.63
10500
31.50
DEPSO [57]
121,412.56
121,419.31
121,468.25
10500
–
HCRO-DE [24]
121,412.55
121,413.11
121,415.66
10500
7.64
OIWO [68]
121,412.54
–
–
10500
–
MABC [33]
121,412.54
–
–
10500
–
ORCSA [69]
121,412.535
121,472.45
121,596.18
10500
3.02
MsBBO [55]
121,412.5344
121417.1877
121450.0026
10500
–
MINLP [70]
121,412.53
121,412.53
121,412.53
10500
39.33
PSO
121,426.21
121,433.14
121,438.85
10500
15.22
MPSO
121,412.33
121,415.65
121,416.73
10500
10.37
OPSO
121,410.32
121,410.55
121,410.86
10500
6.32
Convergence characteristics for test system 3 with 40 generatorsComparison graph for forty generating units with other techniquesScheduled generation among 10 generators with multiple fuel types for minimum cost
Unit
Fuel type
Power output (MW)
P1 (MW)
2
218.722
P2 (MW)
1
211.361
P3 (MW)
1
281.682
P4 (MW)
3
239.379
P5 (MW)
1
279.198
P6 (MW)
3
239.362
P7 (MW)
1
287.796
P8 (MW)
3
239.703
P9 (MW)
3
426.803
P10 (MW)
1
275.994
Total output power
2700 MW
Generation cost ($/h)
623.542
Case 4: Test system 4 for ten generating units with multiple fuels as input
In this case, the performance is evaluated on a system with 10 generating units with multiple fuel options and valve point loading effects. The input data have been referred to from [71]. From the input data, it has been observed that the first generator is having options of two types of fuel and the other generating units have an option of three types of fuel. The total load demand is 2700 MW with no transmission losses. The optimum cost produced during the experiment is 623.542 $/h for OPSO. The comparison of cost with different techniques is represented in Tab. 7 and, it is found that the OPSO is optimizing the system cost for multiple fuel systems. Fig. 9 shows the convergence graph of OPSO and PSO with a faster convergence rate. The scheduled output with different fuels is depicted in Tab. 6.
Comparison of cost with deviation for 10 generating units with different types of fuel
Techniques
Best cost ($/h)
Average cost ($/h)
Worst cost ($/h)
IGA-MU [71]
624.72
627.61
633.87
CGA-MU [71]
624.52
625.87
630.87
RCGA [72]
623.83
623.85
623.89
CBPSO-RVM [73]
623.96
624.08
624.29
ARCGA [74]
623.83
623.84
623.86
NPSO-LRS [48]
624.13
625.00
627.00
DEPSO [57]
623.83
623.90
624.08
PSO
625.21
626.32
627.74
MPSO
624.14
624.89
625.37
OPSO
623.542
623.65
623.33
Comparison graph for ten generating units for case 4
Recent works presented in [75–78] depict interesting optimization works in different domains.
Conclusion
The increasing complexity of today’s electrical networks in Industry 4.0 further adds to the severity of the issue which can be mitigated through robust economic load dispatch strategies. Application of Oscillatory Particle Swarm Optimization algorithm-a meta-heuristic algorithm, to solve complex ELD problems is presented in this paper. The performance of OPSO is evaluated for four different test systems with increasing complications considering various practical technical constraints such as valve-point loading effect, prohibited operating zones, ramp rates, and multiple fuel system. The effectiveness of different techniques for optimizing the cost and their convergence profiles and times have been studied for all these cases. A comparison is performed between the proposed and existing techniques based on the above-discussed problem. It is concluded from the work that the proposed OPSO algorithm provides better performance for minimizing the cost of the power supply, losses, and the convergence time as compared to the existing optimization techniques.
The applied technique can be further applied to an enhanced version of economic dispatch problems such as economic emission dispatch problem, dynamic dispatch problem, and economic dispatch incorporating renewable energy system.
The authors are grateful to the Raytheon Chair for Systems Engineering for funding. The authors are also grateful to the management of authors’ institutions.
Funding Statement: The authors are grateful to the Raytheon Chair for Systems Engineering for funding.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
ReferencesA. J.Wood, B. F.Wollenberg and G. B.Sheblé, J. S.Dhillon and D. P.Kothari, B. H.Chowdhury and S.Rahman, “A review of recent advances in economic dispatch,” G. F.Reid and L.Hasdorff, “Economic dispatch using quadratic programming,” G. B.Sheble, “Real-time economic dispatch and reserve allocate,ion using merit order loading and linear programming rules,” D.Streiffert, “Multi-area economic dispatch with tie line constraints,” Z.Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints,” D. C.Walters and G. B.Sheble, “Genetic algorithm solution of economic dispatch with valve point loading,” N.Noman and H.Iba, “Differential evolution for economic load dispatch problems,” N.Ghorbani and E.Babaei, “Exchange market algorithm for economic load dispatch,” J. Q.James and V.Li, “A social spider algorithm for solving the non-convex economic load dispatch problem,” A.Bhattacharya and P.Chattopadhyay, “Biogeography-based optimization for different economic load dispatch problems,” W.Lin, F.Cheng and M.Tsay, “An improved tabu search for economic dispatch with multiple minima,” L.Han, C. E.Romero and Z.Yao, “Economic dispatch optimization algorithm based on particle diffusion,” S.Hemamalini and S. P.Simon, “Artificial bee colony algorithm for economic load dispatch problem with non-smooth cost functions,” M.Pradhan, P.Roy and T.Pal, “Grey wolf optimization applied to economic load dispatch problems,” V.Kamboj, S. K.Bath and J. S.Dhillon, “Solution of non-convex economic load dispatch problem using grey wolf optimizer,” G.Dhiman, S.Guo and S.Kaur, “ED-SHO: A framework for solving nonlinear economic load power dispatch problem using spotted hyena optimizer,” A.Bhattacharya and P.Chattopadhyay, “Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch,” L.Wang and L.Li, “An effective differential harmony search algorithm for the solving non-convex economic load dispatch problems,” A.Trivedi, D.Srinivasan, S.Biswas and T.Reindl, “Hybridizing genetic algorithm with differential evolution for solving the unit commitment scheduling problem,” V.Karthikeyan, S.Senthilkumar and V. J.Vijayalakshmi, “A new approach to the solution of economic dispatch using particle swarm optimization with simulated annealing,” arXiv preprint arXiv: 1307.3014, 2013.T.Victoire and A.E. Jeyakumar, “Hybrid PSO-SQP for economic dispatch with valve-point effect,” P.Roy, S.Bhui and C.Paul, “Solution of economic load dispatch using hybrid chemical reaction optimization approach,” G.Dhiman, “MOSHEPO: A hybrid multi-objective approach to solve economic load dispatch and micro grid problems,” M. P.Varghese and A.Amudha, “Enhancing the efficiency of wind power using hybrid fire fly and genetic algorithm-economic load dispatch model,” P.Subbaraj, R.Rengaraj and S.Salivahanan, “Enhancement of self-adaptive real-coded genetic algorithm using Taguchi method for economic dispatch problem,” S. C.Dos, Leandro and V.C. Mariani, “An improved harmony search algorithm for power economic load dispatch,” S.Sayah and K.Zehar, “Modified differential evolution algorithm for optimal power flow with non-smooth cost functions,” V.Hosseinnezhad, M.Rafiee, M.Ahmadian and M. T.Ameli, “Species-based quantum particle swarm optimization for economic load dispatch,” S. S.Subramani and P. R.Rajeswari, “A modified particle swarm optimization for economic dispatch problems with non-smooth cost functions,” H.Liu, J.Qu and Y.Li, “The economic dispatch of wind integrated power system based on an improved differential evolution algorithm,” D. C.Secui, “A new modified artificial bee colony algorithm for the economic dispatch problem,” I. A.Farhat and M. E.El-Hawary, “Modified bacterial foraging algorithm for optimum economic dispatch,” in Proc. 2009 IEEE Electrical Power & Energy Conf., Montreal, QC, Canada, IEEE, pp. 1–6, 2009. V. R.Pandi, B. K.Panigrahi, A.Mohapatra and M.Mallick, “Economic load dispatch solution by improved harmony search with wavelet mutation,” G.Dhiman, P.Singh, H.Kaur and R.Maini, “DHIMAN: A novel algorithm for economic dispatch problem based on optimization met Hodus Ing Monte Carlo simulation and astrophysics concepts,” L.Ravi, V.Subramaniyaswamy, V.Vijayakumar, R. H.Jhaveri and J.Shah, “Hybrid user clustering-based travel planning system for personalized point-of-interest recommendation,” in Proc. Int. Conf. on Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Ahmedabad, India, 2020. R. H.Jhaveri, R.Tan and S. V.Ramani, “Real-time QoS routing scheme in SDN-based robotic cyber-physical systems,” in Proc. 5th Int. Conf. on Mechatronics System and Robots, Singapore, pp. 18–23, 2019. S.Kaboli and A.Alqallaf, “Solving non-convex economic load dispatch problem via artificial cooperative search algorithm,” D. N.Vo and W.Ongsakul, “Economic dispatch with multiple fuel types by enhanced augmented Lagrange Hopfield network,” J.Kennedy and R.Eberhart, “PSO optimization,” in Proc. IEEE Int. Conf. Neural Networks, Piscataway, NJ, IEEE Service Center, vol. 4, pp. 1941–1948, 1995. A.Mahor, V.Prasad and S.Rangnekar, “Economic dispatch using particle swarm optimization: A review,” M.Mostafa, S.Kaboli, E.Taslimi-Renani and N.Rahim, “Backtracking search algorithm for solving economic dispatch problems with valve-point effects and multiple fuel options,” A.Rathinam and R.Phukan, “Solution to economic load dispatch problem based on firefly algorithm and its comparison with BFO, CBFO-S and CBFO-Hybrid,” in Proc. Int. Conf. on Swarm, Evolutionary, and Memetic Computing, Berlin, Heidelberg, Springer, pp. 57–65, 2012. K.Bhattacharjee, A.Bhattacharya and S.Dey, “Chemical reaction optimisation for different economic dispatch problems,” T.Ding, R.Bo, F.Li and H.Sun, “A bi-level branch and bound method for economic dispatch with disjoint prohibited zones considering network losses,” S.Özyön and D.Aydin, “Incremental artificial bee colony with local search to economic dispatch problem with ramp rate limits and prohibited operating zones,” A.Selvakumar and K.Thanushkodi, “A new particle swarm optimization solution to nonconvex economic dispatch problems,” V.Hosseinnezhad and E.Babaei, “Economic load dispatch using θ-PSO,” H.Barati and M.Sadeghi, “An efficient hybrid MPSO-GA algorithm for solving non-smooth/non-convex economic dispatch problem with practical constraints,” E.Afzalan and M.Joorabian, “An improved cuckoo search algorithm for power economic load dispatch,” B. K.Panigrahi, V. R.Pandi and S.Das, “Adaptive particle swarm optimization approach for static and dynamic economic load dispatch,” Y.Labbi, D.Attous, H.Gabbar, B.Mahdad and A.Zidan, “A new rooted tree optimization algorithm for economic dispatch with valve-point effect,” K.Bhattacharjee, A.Bhattacharya and S.Dey, “Oppositional real coded chemical reaction optimization for different economic dispatch problems,” G.Xiong, D.Shi and X.Duan, “Multi-strategy ensemble biogeography-based optimization for economic dispatch problems,” G.Xiong and D.Shi, “Orthogonal learning competitive swarm optimizer for economic dispatch problems,” S.Sayah and A.Hamouda, “A hybrid differential evolution algorithm based on particle swarm optimization for nonconvex economic dispatch problems,” G.Binetti, A.Davoudi, F. L.Lewis, D.Naso and B.Turchiano, “Distributed consensus-based economic dispatch with transmission losses,” M.Basu, “Kinetic gas molecule optimization for nonconvex economic dispatch problem,” M. A.Elhameed and A. A.El-Fergany, “Water cycle algorithm-based economic dispatcher for sequential and simultaneous objectives including practical constraints,” J.Cai, Q.Li, L.Li, H.Peng and Y.Yang, “A hybrid CPSO-SQP method for economic dispatch considering the valve-point effects,” G.Binetti, A.Davoudi, D.Naso, B.Turchiano and F. L.Lewis, “A distributed auction-based algorithm for the nonconvex economic dispatch problem,” M. A.Al-Betar, M. A.Awadallah, A. T.Khader and A. L.Bolaji, “Tournament-based harmony search algorithm for non-convex economic load dispatch problem,” G.Chen and X.Ding, “Optimal economic dispatch with valve loading effect using self-adaptive firefly algorithm,” J.Neto, G.Reynoso-Meza, T. H.Ruppel, V. C.Mariani and L. S.Coelho, “Solving non-smooth economic dispatch by a new combination of continuous GRASP algorithm and differential evolution,” M.Azmi, M. A.Awadallah, A. T.Khader, A. L.Bolaji and A.Almomani, “Economic load dispatch problems with valve-point loading using natural updated harmony search,” W.Elsayed and E. F.El-Saadany, “A fully decentralized approach for solving the economic dispatch problem,” A. K.Barisal and R. C.Prusty, “Large scale economic dispatch of power systems using oppositional invasive weed optimization,” T. T.Nguyen and D. N.Vo, “The application of one rank cuckoo search algorithm for solving economic load dispatch problems,” A.Rabiee, B.Mohammadi-Ivatloo and M.Moradi-Dalvand, “Fast dynamic economic power dispatch problems solution via optimality condition decomposition,” C.Chiang, “Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels,” N.Amjady and H.Nasiri-Rad, “Economic dispatch using an efficient real-coded genetic algorithm,” H.Lu, P.Sriyanyong, Y. H.Song and T.Dillon, “Experimental study of a new hybrid PSO with mutation for economic dispatch with non-smooth cost function,” N.Amjady and H.Nasiri-Rad, “Solution of nonconvex and nonsmooth economic dispatch by a new adaptive real coded genetic algorithm,” S. P.RM, P. K.Maddikunta, M.Parimala, S.Koppu S, T. R.Gadekalluet al., “An effective feature engineering for DNN using hybrid PCA-GWO for intrusion detection in IoMT architecture,” T.R.Gadekallu, D. S.Rajput, P. K.Reddy, K.Lakshmanna, S.Bhattacharyaet al., “A novel PCA-whale optimization-based deep neural network model for classification of tomato plant diseases using GPU,” M.Alazab, S.Khan, S. S. R.Krishnan, Q. V.Pham, P. K.Reddyet al., “A multidirectional LSTM model for predicting the stability of a smart grid,” T.Reddy, S.Bhattacharya, P. K.R. Maddikunta, S.Hakak, W. Z.Khanetet al., “Antlion re-sampling based deep neural network model for classification of imbalanced multimodal stroke dataset,”