Generation dispatch of combined solar thermal ... - Springer Link

Computing (2017) 99:59–80 DOI 10.1007/s00607-016-0514-9

Generation dispatch of combined solar thermal systems using dragonfly algorithm Velamuri Suresh1

· S. Sreejith1

Received: 27 December 2015 / Accepted: 2 September 2016 / Published online: 16 September 2016 © Springer-Verlag Wien 2016

Abstract This paper presents a method to solve Static Economic dispatch incorporating solar energy using Dragonfly algorithm. Economic dispatch is carried out considering valve point loading and prohibited operating zone constraints. Solar energy system is modelled using Beta distribution function and included in the objective function. The output power of solar farm is forecasted for four seasons. Different loading conditions are assumed in various seasons for the detailed analysis. Dragonfly algorithm, an emerging optimization technique for constrained optimization problems is applied as the optimization tool here. The proposed method is validated in 6 generator, 15 generator and 17 generator practical south Indian test systems. The results are compared with the state of art heuristic optimization method available in the literature. Keywords Economic Dispatch (ED) · Dragonfly algorithm (DFA) · Beta distribution function · Optimization · Probability density function Mathematics Subject Classification 47N10

1 Introduction In the operation and planning of modern power systems, Economic dispatch (ED) is one of the fundamental issue. ED problem involves scheduling of generating units corresponding to the demand at the cheapest possible cost. Proper ED not only reduces the operating cost, but also improves the reliability and security of the power system. With the continuous rise in demand, generation also increases which aids the ED prob-

B 1

Velamuri Suresh [email protected] School of Electrical Engineering, VIT University, Vellore 632 014, Tamil Nadu, India

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60

V. Suresh, S. Sreejith

lem in becoming highly non linear. Several constraints are imposed on the scheduling of the generators which makes the ED discontinuous function. Penetration of renewable energy sources in to the existing power system has become a trend, due to the falling costs and also depletion of fossil fuels. Emission due to green house gases and many other environmental issues are optimised with the increased usage of renewable energy. Wind and Solar energy sources are the main contributors of this green energy. Excluding the investment and maintenance costs these are the cheapest forms of energy which are abundantly available. The main objective of this paper is to incorporate solar energy in to the traditional economic dispatch and examine the problem in terms of numerical explanation. In this paper both thermal and solar energy sources are simultaneously considered and the fuel cost is minimised. Due to the stochastic nature of solar energy, beta distribution function is used to model the solar farm. Effect of solar energy in terms of operating cost, incurred losses and system generation scheduling are discussed in the paper. In recent years evolutionary algorithms are extensively used for solving nonlinear complex optimization problems. These algorithms are developed based on the behaviour of many living creatures, biological factors and similar other processes. These algorithms were applied on engineering problems like Economic Dispatch (ED), Optimal Power flow (OPF) and Unit Commitment (UC). A brief review of these techniques applied to Economic Dispatch is presented here. We focused on the methods used for handling ED problem rather than giving detailed summary. Convectional ED had been solved using many classical, analytical and meta heuristic methods. Due to the nonlinearities, classical methods like gradient method (GM) [1], Lambda iteration method (LIM) [2], Linear programming (LP) [3] and Quadratic Programming(QP) [4] failed in constraint handling effectively. These methods involves high computational time and large number of iterations. Even though meta heuristic techniques does not guarantee global optimal solution, they are extensively used due to their ability in delivering a sub optimal solution in quick computational time. Methods like Genetic Algorithm (GA) [23], Particle swarm optimization (PSO) [23], Hybrid Particle swarm optimization (HPSO) [5], Cuckoo search (CS) [6], Krill herd Algorithm (KHA) [7], Ant colony optimization (ACO) [8], Particle Diffusion (PD) [9], Firefly Algorithm (FA) [10], Hybrid imperialist competitive-sequential quadratic programming (HIC-SQP) [11], Distributed Sobol PSO and TSA (DSPSO-TSA) [12], Modified harmony search (MHSA) [13], Immune Algorithm (IA) [14], Biogeography based optimization (BBO) [15], Improved Coordinated Aggregation-Based PSO (ICA-PSO) [16], Improved Harmony search (IHS) [17], SA-PSO [18], Artificial Immune system (AIS) [19], Differential Evolution (DE) [20], Modified Artificial bee colony (MABC) [21], Differential Harmony search (DHS) [22], New PSO with Local Random Search (NPSO-LRS) [24], Modified Tabu search (MTS) [25], Simulated Annealing (SA) [25], Modified Differential Evolution(MDE) [27], Bacterial Foraging with Nelder Mean Algorithm (BFO-NM) [28], Evolutionary strategy optimization (ESO) [29], Adaptive PSO (APSO) [41], Hybrid Differential Evolution With Biogeography-Based Optimization (HDEBBO) [42], Self-Organizing Hierarchical Particle Swarm Optimization (SOH-PSO) [43] are used for solving ED by considering the Valve point effect, Prohibited operating zones (POZ), Emission and Ramp rate.

123

Generation dispatch of combined solar thermal systems. . .

61

Recently some papers solved ED considering renewable energy sources in addition to existing thermal units. In [30] real time ED considering both solar and wind powered generators is solved. In [31] Combined economic emission dispatch considering solar power is solved for both static and dynamic conditions. Dynamic ED considering renewable energy sources is solved using WAPSO in [32]. In [33] combined wind thermal economic dispatch is presented. Dynamic feed in tariff cost function in contrast to feed in tariff is introduced. In paper [34] the wait and see approach of probabilistic wind is discussed and it is incorporated with existing system. Market clearing mechanism is proposed for combined wind thermal system in [35]. Uncertainties in both wind velocity and load demand are modelled and incorporated in the system. In [36] combined economic dispatch considering renewable sources and plug in vehicles with cost function for wind is introduced. Paper [37] introduces the concept of spinning reserve in wind thermal systems and CMA-ES with mean learning technique is used for solving economic dispatch. In [39] multi objective economic emission dispatch considering wind power is solved using point estimate method. In [40] combined wind solar and thermal units are considered for the dispatch. Based on the variability of renewable energy sources reserve risk is also included in the model for improving the economy of the system. From the above discussed methods, it is obvious to say that many optimization methods has been applied till date for solving the convectional ED problem and produced promising results. Even though they produce better results, methods like GA, PSO, SA-PSO has limitation in terms of computational time and number of evaluations for solving the problem. PSO even though is very simple in implementation it suffers from locking at a local minimum. In solving ED problem DE, BBO, IHS requires large number of evaluations which leads to high computational time. From [49,50] it is observed that in AgGGSA and GWO exploration and exploitation are balanced the computational time becomes slow when applied to medium size power systems. With the addition of renewables, the complexity of the problems increases even more compared to the convectional case. Solving this problem requires a robust optimization for obtaining global minimum in less computational time and number of evaluations. Dragonfly algorithm, a recent optimization which over comes the above said problems in case of economic dispatch is presented in this paper. It is also observed that, in modelling the renewable energy sources, many authors do not consider the past data for estimation of the output. They considered a basic design that is commonly applied. Modelling the renewable energy sources based on historical data according to the seasons gives us the exact view of the problem and helps in minimising the errors like over estimation and under estimation of output.

1.1 Proposed work In this work, a new method of dispatching in accordance with the seasonal estimation of solar power is presented. Combined solar thermal Economic Dispatch considering seasonality, prohibited operating zones is solved using a novel optimization technique. For validation the method is first tested on existing 6, 15 generator test systems and compared with many other algorithms stated in literature. Then the proposed method

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62


is applied on real time South Indian utility 89 bus test system for the cases without and with integration of solar energy. The results proved that the developed model is efficient in handling the demand generation coordination with incorporation of renewable energy sources. The proposed algorithm is efficient in solving the non linear constrained optimization in very less evaluations and at a faster convergence rate. The remaining paper is organised as follows. In Sect. 2 problem formulation of the Convectional Economic dispatch, Modelling of Solar farm and Static economic dispatch considering solar energy are presented. Section 3 explains the Dragonfly algorithm and its implementation to ED problem. In Sect. 4 simulation results for cases without and with solar farm in solving ED for various test systems are presented and compared with other methods available in literature. Finally Sect. 5 gives the conclusion of the discussions.

2 Problem formulation The main goal of Proposed Solar thermal economic dispatch is to minimise the generation cost of all the participating units and to utilise the available solar power by satisfying all the constraints posed by the generators. 2.1 Fuel cost function and operating constraints for thermal units The quadratic cost function considering the effect of valve point effect is given by (1) F1i = ai Pgi2 + bi Pgi + ci + ei × sin f i × Pimin − Pi Economic dispatch for thermal units is given by C1 = min

n

F1i

(2)

i=1

where F1i is the fuel cost of thermal generators, ai , bi , ci , ei , f i are the cost coefficients of ith thermal generator. The objective function is subjected to following constraints. Operating limits The generation should be within its maximum and minimum limits Pgimin ≤ Pgi ≤ Pgimax

(3)

where Pgimin , Pgimax are the lower and upper bounds for ith thermal generator. Prohibited operating zones ⎫ ⎧ lb min ⎪ ⎬ ⎨ Pgi ≤ Pgi ≤ Pi,1 ⎪ ub lb ≤ Pgi ≤ Pi,k , k = 2, 3, . . . , Z i POZ Pgi ∈ Pi,k−1 ⎪ ⎭ ⎩ P ub ≤ Pgi ≤ P max ⎪ i,z i gi

123

(4)


63

lb , P ub are the lower and upper where k is the number of POZ’s for ith generator, Pi,k i,z i limits of the kth POZ.

Transmission power loss The transmission power loss is calculated using Krone’s reduction method n n PL = Pi Bi j P j + B0i + B00 (5) i=1 j=1

i∈ j

where Bi j , B0i , B00 are the loss coefficients of the generator. Power balance constraint The total generation should be equal to the sum of the total demand and losses n Pgi = Pd + PL (6) i=1

where Pgi , Pd are the power generation of ith unit and Power demand respectively. 2.2 Solar farm modelling For characterising the Stochastic nature of renewable energy sources probability density function is used. Beta distribution function is the best suited PDF in case of solar irradiance modelling compared to others [48]. Modelling of Solar power using PDF requires historical irradiance data of the site chosen. 2.2.1 Details of the resources Vellore location, Tamilnadu, India (79.15◦ E, 12.95◦ N) has plentiful of Solar irradiance throughout the year. So irradiance data from this location is chosen for modelling Solar farm. The seasonal changes occurred for this location are processed and sub divided into four seasons. In all the seasons, each day is sub divided in to 24 segments which represents the hours of a day. Data of past 1 year is taken from [44] and using this data, parameters of the Beta distribution function are calculated. 2.2.2 Solar energy system modelling Solar irradiance distribution is modelled using Beta distribution function [45] and is written as follows f p (s) =

(α + β) (α−1) s (1 − s)(β−1) , 0 ≤ s ≤ 1 α, β ≥ 0 (α)(β)

(7)

μβ where β = (1 − μ) μ(1+μ) − 1 and α = 1−μ . 2 σ where f p (s), (.), s are the beta distribution function, gamma function and random variable of solar irradiance (kw/m2 ) respectively. μ, σ are the mean and standard deviation obtained from the historical data.

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64


The power output from the solar panel is obtained using the following equation P(s) = Po (s) ∗ f p (s)

(8)

The total PV panel power output in a specified time is given by

1

TP =

Po (s) ∗ f p (s)ds

(9)

0

where Po (s) = N ∗ F F ∗ V y ∗ I y

(10)

The V–I characteristics of a PV module are calculated as follows FF =

VM P P T ∗ I M P P T VOC ∗ I SC

(11)

Vy = Voc − K v ∗ Tcy

(12)

I y = s [Isc + K i (Tcy − 25)] N O T − 20 Tcy = T A + s 0.8

(13) (14)

The specifications of the 220 W PV panel which is used for modelling is taken from [46]. 2.3 Fuel cost and operating constraints for solar farm The fuel cost of solar farm is given by F2i = Tsi × Psi

(15)

where F2i , Tsi , Psi are the fuel cost, tariff and power generated by ith solar farm. Economic dispatch of solar farm is given by C2 = min

u

F2i

(16)

i=1

Constraint for solving ED with solar farm is u

Psi ≤ 0.3 × Pd

i=1

where Psi , Pd are the solar power generated and power demand respectively.

123

(17)


65

2.4 Proposed objective function In the proposed work the solar farm is modelled and the output is entirely utilised if it satisfies the constraint in Eq. (17). The updated objective function is given by C = C1 + C2 = min

n

F1i + min

i=1

u

F2i

(18)

i=1

The power balance constraint is updated as follows n i=1

Pgi +

u

Psi = Pd + PL

(19)

i=1

3 Dragonfly algorithm Dragonfly algorithm (DFA) is proposed by Seyedali Mirajalli in 2015. This technique is based on the swarming behaviour of dragonflies. Here, the parameters of exploration and exploitation are modelled carefully so as to obtain global optimization. Dragonflies are basically carnivorous insects which predates other small insects like mosquitoes and butterflies as food. The life cycle of dragonflies has two stages, nymph and metamorphosis. Nymph dragonflies eat marine insects and also small fish by utilising their extreme skill of swimming (swarming). Usually the swarms are both static and dynamic in nature. Static swarm does hunting in which they form a small group and predates small insects confined to a small area. Dynamic swarms form as large groups and they travel in one direction for long distances. The main goal of dragonflies should be attraction to the food source and distraction from an enemy. The three important stages in the swarming behaviour of dragonflies are explained and mathematically shown as follows. 1. Separation (S) In this stage the swarms are separated from other individuals which avoids collision from the neighbours. This separation is given by Si = −

N

X − Xj

(20)

j=1

whereas X and X j are the position of current individual and neighbouring position of jth individual respectively. Number of neighbourhoods are given by N. 2. Alignment (A) Here the velocity of each individual is matched with the other. Alignment is represented by N Ai =

j=1

N

Vj

−X

(21)

where Vj is the velocity of neighbouring individual j.

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66


3. Cohesion (C) This refers to the attraction of the swarm towards the centre of the group of swarms. N j=1 X j −X (22) Ci = N whereas X j is the position of the jth neighbouring individual and X is the position of current individual respectively. 4. Attraction towards the food (F) source is mathematically represented by Fi = X+ − X

(23)

where X + is the food source position and X is the position of current individual. 5. Distraction from the enemy (E) is calculated by Ei = X− + X

(24)

where X − is the enemy position and X is the current individual position. Exploration and exploitation mainly depends on the factors S, A, C, F, E. By proper tuning of these parameters global optimization can be achieved. In the optimization process neighbourhood plays a key role. With the created neighbourhood around each individual the performance of the dragonflies are extremely achieved. The step vector is updated using the following equation X t+1 = (s Si + a Ai + cCi + f Fi + eE i ) + wX t

(25)

The updated position vector is calculated as follows X t+1 = X t + X t+1

(26)

The randomness, stochastic behaviour of the dragonflies can be increased by introducing levy flight in the given search space thereby improving the exploration. The position update of the dragonflies is given by X t+1 = X t + Levy(d) × X t

(27)

where d is the dimension of the search space and t is the current iteration. Levy flight is given by r1 × σ (28) Levy(d) = 0.01 × 1 |r2 | β r1 and r2 are the random numbers in [0,1] and β is a constant and σ is given by ⎞ β1 (1 + β) × sin πβ 2 ⎟ ⎜ ⎠ σ =⎝ β−1 ×β ×2 2 1+β 2 ⎛

123

(29)


67

Fig. 1 Pseudo code of Dragonfly algorithm (adapted from [38])

3.1 Implementation of Dragonfly algorithm for economic dispatch problem The proposed DFA method is more powerful and robust compared to other meta heuristic methods for solving non convex and large scale optimization problems. So it is efficiently applied for the ED problem. The pseudo code for Dragonfly algorithm is shown in Fig. 1. The steps used for solving the ED problem are shown in flow chart in Fig. 2.

4 Results and discussion Static economic dispatch considering solar farm is solved using Dragonfly algorithm. Prohibited operating zones, valve point and ramp rate are considered while solving the problem. The performance of proposed algorithm is tested on three test systems. Four case studies are presented in the following section. In first three cases Static economic dispatch without considering solar farm are presented. Case 4 describes the effects of incorporating solar farm in existing power network.

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Fig. 2 Flowchart for solving economic dispatch using DF algorithm

4.1 Case 1 Here test system consisting of six generators and a demand of 1263 MW is considered. The data of the test system is obtained from [23]. The number of search agents, maximum iterations considered in the DFA optimization are 20 and 100 respectively.

123

173.488

173.239 263.764

263.316

264.176

261.026

262.462

P3

Bold values indicate best-known performance of the problem

446.727

MHSA [13]

172.216

446.676

447.399

IA [14]

BBO [15]

187.787

439.293

DSPSO-TSA [12]

172.205

445.969

DFA

P2

P1

Algorithm

138.832

138.000

143.675

129.497

142.696

P4

Table 1 Analysis of dispatch for 6 generator system with Pd = 1263 MW

165.649

165.410

161.342

171.710

164.857

P5

86.9463

87.0797

87.2039

86.1648

86.7047

P6

1275.40

1275.44

1275.29

1275.51

1274.89

PG

12.4181

12.446

12.2903

13.0421

11.8952

PL

15,442.52

15,443.09

15,442.937

15,441.57

15,436.869

Cost ($/h)

NA

3.25

NA

0.37

0.9287

Time (s)

Generation dispatch of combined solar thermal systems. . . 69

123

123

15,438.49

15,436.87

DFA

15,439.93

15,439.01

15,446.22

15,444.25

15,444.33

–

15,455

15,472

15,449.87

1,5449.89


15,441.57

BBO [15]

DSPSO-TSA [12]

15,443.24

15,443.1

ICA-PSO [16]

HIC-SQP [11]

15,449.03

15,444.3

IHS [17]

15,442.94

15,447

SA-PSO [18]

15,442.52

15,448

AIS [19]

IA [14]

15,449.77

DE [20]

MHSA [13]

15,443.1

15,449.89

MABC [21]

15,449.99

15,452

15,450

15,449.9

NPSO_LRS [24]

15,492

15,453.64

15,469

Maximum

DHS [22]

15,450.06

15,450

MTS [25]

15,459

GA [23]

PSO [23]

Minimum

Algorithm

Table 2 Comparison of best solution in terms of cost ($/h) optimization

15,437.16

15,438.89

15,443.84

15,443.59

15,444.04

15,443.1

15,443.97

15,449.87

15,447

15,459.7

15,449.78

15,449.89

15,449.93

15,450.5

15,454

15,451.17

15,469

Average

0.37679

NA

1.07

NA

1.04109

NA

NA

4.5312

NA

NA

2.04E−02

6.4E−08

14.86

1.29

NA

0.4158

0.2528

Std. dev.

0.9287

NA

0.37

NA

0.796

NA

NA

NA

6.25

0.03

0.01

NA

NA

0.93

NA

NA

7.58

Time (s)

2000

NA

4200

NA

NA

50,000

20,000

100,000

20,000

NA

36,000

NA

3000

20,000

20,000

100,000

20,000

Evaluations

70 V. Suresh, S. Sreejith


71

Table 3 Comparative analysis of proposed method S. no.

Algorithm

Cost ($/h)

Loss (MW)

Gen (MW) 1276.03

1

GA [23]

15,459

13.0217

2

MTS [25]

15,450.06

13.0205

1276

3

PSO [23]

15,450

12.9584

1276.01

4

NPSO_LRS [24]

15,450

12.947

1275

5

DHS [22]

15,449.9

12.959

1275.95 1275.95

6

MABC [21]

15,449.89

12.9582

7

DE [20]

15,449.77

12.957

1275.947

8

AIS [19]

15,448

12.655

1275.655

9

SA-PSO [18]

15,447

12.733

1275.73

10

IHS [17]

15,444.3

12.3918

1275.38

11

ICA-PSO [16]

15,443.24

12.47

1275.47

12

BBO [15]

15,443.1

12.446

1275.44

13

IA [14]

15,442.94

12.2903

1275.219

14

MHSA [13]

15,442.52

12.4181

1275.418

15

DSPSO-TSA [12]

15,441.57

13.0421

1275.514

16

HIC-SQP [11]

15,438.49

12.1

1275.1

17

DFA

15,436.87

11.8952

1274.895


The results obtained are presented in Tables 1, 2 and 3. The results are compared with many recent optimization techniques and proved to be the best in terms of minimum objective function value, convergence time, evaluations and power loss. In Table 1, the dispatch of each generator, power loss and time for convergence obtained using DFA are presented and compared with the recent methods in literature. From the table it is observed that the cost of generation and power loss is very low when compared to other methods. Table 2 gives the minimum, maximum and average values of the cost function obtained after running 50 trails. It is observed that the worst value obtained using DFA is better than minimum value obtained by other methods. The values obtained for 50 trails are sorted and then plotted in Fig. 3. The standard deviation is 0.3769 which indicates that the algorithm outperforms in terms of search ability for the specified parameters. The DFA converges in 0.92 s for 100 iterations. It is also observed that the DFA requires only 2000 evaluations for cost function optimization, which is the lowest of all the methods. In Table 3 generation cost, losses and total generation required are compared with the other methods. From all the observed results it is proved that DFA gives the best values in terms of cost, generation and transmission loss.

4.2 Case 2 In this case 15 generator test system with a demand of 2630 MW is considered. The data for the test system is taken from [14]. Here also the number of search agents and

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72


Fig. 3 Deviation in cost for different trails

Table 4 Analysis of cost and losses in 15 generator test system S. no.

Algorithm

Cost ($/h)

Loss (MW)

Generation (MW)

1

GA [23]

33,113

38.27

2668.4

2

TSA [25]

32,918

33.811

2663.7

3

PSO [23]

32,858

32.43

2662.4

4

AIS [19]

32,854

32.4

2662.04

5

SA [25]

32,786

33.27

2663.29

6

BFO [28]

32,784.5

28.947

2658.947

7

TS [25]

32,762

31.41

2661.53

8

MTS [25]

32,716

31.35

2661.36

9

DSPSO-TSA [12]

32,715

30.952

2660.96

10

SA-PSO [18]

32,708

30.908

2630

11

CCPSO [26]

32,704

30.6616

2660.66

12

MDE [27]

32,704

30.62

2660.62

13

IA [14]

32,698

30.0187

2660.019

14

DE [20]

32,588.87

27.975

2657.96

15

ESO [29]

32,568.54

23.85

2653.85

16

DFA

32,548.4

27.016

2657.016


maximum iterations in the DFA optimization are 20 and 100 respectively. Economic dispatch considering prohibited operating zones and valve point is conducted using DFA algorithm and the results are furnished in Tables 4 and 5 for comparison in terms of cost and losses.

123


73

Table 5 Analysis of proposed method for 15 generator test system Unit

DFA

DE [20]

IA [14]

DSPSO-TSA [12]

AIS [19]

1

455

454.997

455

453.627

441.159

2

455

419.997

379.9999

379.895

409.587

3

130

129.997

130

129.482

117.298

4

130

129.998

129.9999

129.923

131.258

5

241.6315

269.917

169.9999

168.956

151.011

6

460

459.99

459.9999

45.9907

466.258

7

465

429.995

429.9999

42.9971

423.368

8

60

60.007

67.9628

103.673

99.948

9

25

25.001

65.7269

34.909

110.684

10

33.3856

63.111

156.3294

154.593

100.229

11

66.9989

79.973

80

79.559

32.057

12

80

79.983

79.9999

79.388

78.815

13

25

25.001

25

25.487

23.568

14

15

15.001

15

15.952

40.258

15

15

15

15

15.64

36.906

TP

2657.016

2657.966

2660.019

2660.96

2662.04

PL

27.016

27.975

30.0187

30.952

32.4075

TC

32,548.4

32,588.87

32,698.2

32,715.06

32,854


In Table 4 the total generation cost, transmission loss, and power generation obtained using DFA are compared with other available methods in literature. The total power generation to meet the demand using proposed method is 2657.016 MW. In the proposed method the total generation cost obtained is 32548 $/h. Even though the transmission loss obtained by ESO [12] method is less than the proposed method, the cost of generation is high. Table 5 gives a comparison of the dispatch by each generator and power loss. The convergence curve plotted for proposed method is shown in Fig. 4. It is observed from the plot that the minimum cost is obtained by the proposed method at a faster rate in short computational time of 1.2 s. 4.3 Case 3 The significance of proposed algorithm is presented here by considering a practical test system namely South Indian 86 bus test system, located in India. This test system consists of 17 generators, 86 buses and 131 lines. The power demand is 1796.3 MW for the particular hour of dispatch. The data for the test system is taken from [46]. The number of search agents and maximum iterations considered are 20 and 100 respectively for the proposed method. Transmission loss and constraints like ramp rate, up reserve and down reserve are considered while solving the dispatch problem. The obtained results are compared with PSO, ABC methods and furnished in Table 6. From the table it is inferred that optimal cost is obtained using DFA method is 82,967.55 $/h which is very much less compared to PSO and ABC methods.

123

74


Fig. 4 Convergence curve while solving cost function for 15 generator test system

Table 6 Comparison of dispatch for south Indian 86 bus test system


123

Unit

PSO

ABC

DFA

1

209.971

209.3704

210

2

210

200.6843

210

3

65.9786

79.1516

47.1845

4

117.7057

197.7156

210

5

167.729

186.7303

148.2682

6

78.1428

36.001

136.6314

7

198.9127

210

210

8

12.3993

33.5684

13.2471

9

26.798

46.3405

38.6956

10

84.2792

21.3527

10

11

11.2227

68.3076

10

12

10

22.4827

10

13

186.6047

117.9939

210

14

93.2971

20

87.059

15

101.4586

114.9714

66.976

16

125.7389

154.0218

210

17

143.635

126.3417

20

TG (MW)

1843.873

1845.034

1848.062

PL (MW)

47.5733

48.734

51.7618

Cost ($/h)

83,637

83,571

82,967.55

0

0

0

0

0

0

0.20136

0.431697

0.663303

0.7296

0.74

0.7891

0.7691

0.769315

0.584348

0.361169

0.136045

0

0

0

0

0

0

0

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

0

0

0

0

0

0

0

0.032172

0.079165

0.107052

0.116533

0.178971

0.116533

0.116533

0.076632

0.1121

0.070033

0.055999

0

0

0

0

0

0

0

0

0

0

0

0

0.002793

0.111815

0.296098

0.472228

0.632141

0.76663

0.853043

0.844174

0.763489

0.613065

0.42612

0.222196

0

0

0

0

0

0

Mean

Mean

Std. dev.

Summer

Spring

1

Hour

0

0

0

0

0

0

0.001947

0.047759

0.081885

0.133641

0.153182

0.163109

0.146666

0.14108

0.130652

0.125786

0.094075

0.056958

0

0

0

0

0

0

Std. dev.

Table 7 Mean and standard deviation for various seasons of a typical day in a year

0

0

0

0

0

0

0.000522

0.061522

0.225337

0.405446

0.577348

0.728913

0.782141

0.783196

0.701761

0.581033

0.3945

0.202283

0

0

0

0

0

0

Mean

Autumn

0

0

0

0

0

0

0.001172

0.045853

0.093037

0.14775

0.186095

0.172245

0.182147

0.182389

0.180827

0.136235

0.100378

0.051592

0

0

0

0

0

0

Std. dev.

0

0

0

0

0

0

0

0.035163

0.208533

0.389033

0.541826

0.643511

0.695783

0.710598

0.653511

0.528457

0.339957

0.15412

0

0

0

0

0

0

Mean

Winter

0

0

0

0

0

0

0

0.036221

0.086609

0.137478

0.170613

0.212183

0.209585

0.190739

0.158499

0.127166

0.087622

0.042166

0

0

0

0

0

0

Std. dev.


123

76


Fig. 5 Probability of solar output for various seasons in an year

4.4 Case 4 In the south Indian 86 bus test system location named Vellore is chosen for incorporating solar farm. An installed capacity 220 MW is considered for the purpose. Four different demands are assumed in the considered four seasons based on the load data available in [47]. The probability of solar power output is calculated using beta distribution function as explained in Sect. 2.2. The preliminaries for calculating this beta distribution function are the mean and standard deviation obtained from the historical data. The mean and standard deviation for various seasons in a year are calculated and tabulated in Table 7. The variation of probability of solar irradiance is shown in Fig. 5. Based on the estimated power output from the solar farm and load demand, the combined static economic dispatch is solved. For the study Solar farm output at 12.00◦ clock of the day is chosen. The results are presented by considering that obtained solar energy has been purchased from the private utility at a fixed feed in tariff of 40 $/MW. Cases without and with solar farm are compared in all the seasons and presented in Table 8. It is observed from the table, that there are net savings in each season based on the available solar power and load demand. The cost comparison with out and with solar farm in South Indian utility system is shown in Fig. 6. It is observed that highest saving are obtained in Spring and lowest profit is obtained in Autumn. This profit completely depends on the availability of Solar energy for the particular hour of the day. With the increase in penetration of solar energy there is considerable reduction in cost as well as loss compared to the base case is ascertained. From the analysis it is inferred that incorporation of solar farm with the existing thermal units has a significant impact in terms of cost minimisation and losses. If the

123

2119.78

–

2119.78

96,006.24

–

96,006.24

1812.94

Pg thermal

Pg solar

Total Pg

Fuel cost

Solar cost

Total cost

Net saving

2050

69.7804

Loss

94,193.3

6196

87,997.3

2119.041

154.9

1964.141

69.041

2050

1496.3807

88,285.31

–

88,285.31

1958.682

–

1958.682

58.682

1900

Autumn Without

With

Summer

Without

Demand

Season

86,788.93

5717.6

81,071.33

1952.541

142.94

1809.601

52.5413

1900

With

Table 8 Profit analysis on South Indian 86 bus system without and with solar farm

1732.8812

78,595.2

–

78,595.2

1752.163

–

1752.163

52.1632

1700

Without

Winter

76,862.31

5128.4

71,733.91

1747.734

128.21

1619.524

47.7338

1700

With

1829.9452

93,615.7

–

93,615.7

2076.705

–

2076.705

76.7046

2000

Without

Spring

91,785.76

5632.8

86,152.95

2066.66

140.82

1925.84

66.6633

2000

With


123

78


Fig. 6 Profit analysis for various seasons without and with solar farm

solar farm is owned by the power producers then huge savings can be expected as the solar energy is almost free.

5 Conclusion In this paper static economic dispatch with incorporation of solar energy is solved using Dragonfly algorithm. Practical constraints like valve point effect, ramp rate and prohibited operating zones are considered while solving the constrained optimization. Stochastic nature of solar irradiance is modelled using beta distribution function and incorporated in to the existing test system. Historical data considered the output estimation is processed into four seasons. In each season different loading conditions are considered and simulations are performed. The analysis on 6 and 15 generator systems are compared with the some existing optimization methods. DFA method gives minimum cost, losses and converges in extremely low running time. Cases without solar and with solar energy is carried out on South Indian 17 generator system using DFA for various seasons and the profit analysis is presented. It can be concluded that if there is abundant availability of sun in chosen location more amount of power is obtained. If this obtained solar power is properly utilised, by integrating it with existing grid the economy can be increased and the losses in the system are minimised. The proposed methodology can also be implemented by incorporating any other renewable energy sources and also by including the investment costs for these sources in future works.

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