Optimal Short Term Hydrothermal Scheduling

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Optimal Short Term Hydrothermal Scheduling

Gaurav Kumar1

Department of Electrical Engineering, FOT , UTUCampus,

Ravi Shankar Bahuguna2 Department of Electrical Engineering, JBIT Dehradun,

Lakhan Singp

Department of Electrical Engineering, JBIT Dehradun,

Abstract In this paper, short term hydrothermal scheduling is introduced. . The primary objective of the short term hydrothermal scheduling problem is to determine the optimal generation schedule of the thermal and hydro units to minimize the total operation cost of the system over the scheduling time horizon (typically one day) subjected to a variety of thermal and hydraulic constraints. The hydrothermal generation scheduling is mainly concerned with both hydro unit scheduling and thermal unit dispatching.

Keywords: GENTIC ALGORITHM, LAGRANGE'S THEOREM.

I. INTRODUCTION

With extensive interconnection of the electric networks, the energy emergency on the planet and nonstop ascent in costs, it is exceptionally fundamental to lessen the running expenses of electric energy. A sparing in the operation of the power framework achieves a noteworthy decrease in the working expense and in addition in the amount of fuel expended. The principle point of current electric power utilities is to give fantastic solid power supply to the buyers at the least conceivable cost while working to meet the cutoff

Points and imperatives forced on the creating units and ecological contemplations.

The fundamental here and now hydrothermal scheduling case requires that a given measure of water be utilized as a part of such a path as to limit the cost of running the warm units. In the, here and now hydrothermal scheduling case the warm framework is spoken to by a comparable unit PS as done in the Fig 1 and a hydroelectric plant PH. It is expected that the Hydro-plant is not adequate to supply all the heap requests amid the period and that there is a

Figure: a general outline of a hydro warm plant

most extreme aggregate volume of water that might be released all through the time of T max hour

As hydro creating units don't results any fuel cost the hydrothermal scheduling issue is planned to limit the aggregate cost of warm plant while making utilization of the accessible hydro assets although much as could be expected. The target work and related constraints of the issue are detailed as take after:

Genetic algorithm

Genetic Algorithms (GAs) are based on analogy, and are adaptive heuristic search algorithm based on , evolutionary ideas of natural selection and genetics. As such, they GAs represent an intelligent exploitation of the random search used, to solve search and optimization problems. Although randomized, GAs are by no means random, instead they are exploit historical information to direct the search in to the region of better performance with in the search space. The basic techniques of the GA are designed to simulate processes in natural systems necessary for evolution, especially those follow the principles first laid down by Charles Darwin of , "Survival Of The Fittest". Since in nature, competition among individuals for scanty resources, results in the fittest individuals dominating over the weaker ones.

The basic of genetic algorithm contains breeding process. The breeding process is the heart of the genetic algorithm. It is in this process, the search process creates new and hopefully fitter individuals.

The breeding cycle consists of three steps:

  1. Selecting parents.

  2. Crossing the parents to create new individuals (offspring or children).

  3. Replacing old individuals in the population with the new ones.

    Selection

    Selection is the process of choosing two parents from the population for crossing. After deciding on an encoding, the next step is to decide how to perform selection i.e., how to choose individuals in the population that will create offspring for the next generation and how many offspring each will create

    Figure: Breeding Cycle

    Crossover (Recombination)

    Crossover is the process of taking two parent solutions and producing from thema child. After the selection (reproduction) process, the population is enriched with better individuals. Reproduction makes clones of good strings but does not create new ones. Crossover operator is applied to the mating pool with the hope that it creates a better offspring.

    Mutation

    After crossover, the strings are subjected to mutation. Mutation prevents the algorithm to be trapped in a local minimum. Mutation plays the role of recovering the lost genetic materials as well as for randomly disturbing genetic information. It is an insurance policy against the irreversible loss of genetic material. Mutation has traditionally considered as a simple search operator.

    Figure: Genetic Algorithm

    If crossover is supposed to exploit the current solution to find better ones, mutation is supposed to help for the exploration of the whole search space. Mutation is viewed as a background operator to maintain genetic diversity in the population. It introduces new genetic structures in the population by randomly modifying some of its building blocks. Mutation helps escape from local minimas trap and maintains diversity in the population. It also keeps the gene pool well stocked, and thus ensuring ergodicity. A search space is said to be ergodic if there is a non-zero probability of generating any solution from any population state.

    There are many different forms of mutation for the different kinds of representation. For binary representation, a simple mutation can consist in inverting the value of each gene with a small probability. The probability is usually taken about 1/L, where L is the length of the chromosome. It is also possible to implement kind of hill-climbing mutation operators that do mutation only if it improves the quality of the solution. Such an operator can accelerate the search. But care should be taken, because it might also reduce the diversity in the population and makes the algorithm converge toward some local optima. Mutation of a bit involves flipping a bit, changing 0 to 1 and vice-versa.

    Flipping

    Flipping of a bit involves changing 0 to 1 and 1 to 0 based on a mutation chromosome generated. In mutation-flipping concept a parent is considered and a mutation chromosome is randomly

    Generated. For a 1 in mutation chromosome, the corresponding bit in parent chromosome is flipped (0 to 1 and 1 to 0) and child chromosome is produced. In the above case, there occurs 1 at 3 places of mutation chromosome, the corresponding bits in parent chromosome are flipped and child is generate

    Parent

    1 0 1 1 0 1 0 1

    Mutation chromosome

    1 0 0 0 1 0 0 1

    Child

    0 0 1 1 1 1 0 0

    Figure: Mutation flipping

    Optimization Techniques

    1. Determinism: A purely deterministic search may have an extremely high variance in solution quality because it may soon get stuck in worst case situations from which it is incapable to escape because of its determinism. This can be avoided, but it is a well-known fact that the observation of the worst-case situation is not guaranteed to be possible in general.

    2. Non determinism: A stochastic search method usually does not suffer from the above potential worst case wolf trap phenomenon. It is therefore likely that a search method should be stochastic, but it may well contain a substantial portion of determinism however. In principle it is enough to have as much non determinism as to be able to avoid the worst-case wolf traps

Time

PD

P1

P2

P3

PH N

CRPILETS 20

19 CFouneflecroenstce Proceed

ings

(hr)

(MW)

(MW)

(MW)

(MW)

(MW)

(MW)

Rs/hr

1

175(MW)

84

37

20

38.5

4.58

1.55e+003

2

190(MW)

75.3

37.7

37.6

41.8

2.56

1.60+003

3

220(MW)

65.15

37.71

72.25

48.4

3.5

1.75+003

4

280(MW)

67.15

43.78

114

61.6

2.01

2.08+003

5

320(MW)

89.99

54.92

114

70.4

9.31

2.31+003

6

360(MW)

112.57

65.82

114

79.2

11.59

2.56+003

Local determinism: A purely stochastic method is usually quite slow. It is therefore reasonable to do as much as possible efficient deterministic predictions of the most promising directions of (local) proceedings. This is called local hill climbing or greedy search according to the obvious strategies

Implementation Of Short-Term Hydrothermal Scheduling The short term hydrothermal scheduling problem based on Lagrange Multiplier, simulated annealing and genetic algorithm has been tested on three different test systems. Three different test systems of thermal power plant and one hydro which share 22% of total load demand are taken to study the problem.

    1. Case Study 1: Three Unit System [1]

      1. Lagrange Multiplier Method

        The cost characteristics of the three units are given as F1=0.006P1²+5.506P1+264.634 Rs/hr F2=0.016P2²+5.2P2+154.2 Rs/hr F3=0.005P3²+5.67P3+261.1 Rs/hr

        The unit operating constraints are- 40MW P1 225MW

        20MW P2 240MW

        20MW P3 114MW

        The B matrix of the transmission line loss coefficient is given by

        B=1e-2.*[0.027251 -.003506 -.036788

        -.003506 .030896 -.005653

        -.036788 -.005653 0.32295];

        For the above system considering 24 hours loads

        RESULTS: The result of Lagrange multiplier of short term hydro thermal scheduling shown below table 2 of 24 hour load. The Total Fuel Cost is 6.4557e+004 Rs/Hr

        Lagrange Method

        24 hours fuel cost curve (lagrange multiplier method)

        5000

        FUEL COST (Rs/Hr.)

        FUEL COST (Rs/Hr.)

        4000

        3000

        2000

        1000

        0

        0 5 10 15 20 25

        TIME (HOURS)

        Figure: fuel cost curve for 3 unit system

        Time (hr)

        PD (MW)

        P1 (MW)

        P2 (MW)

        P3 (MW)

        PH (MW)

        PL (MW)

        Fuel cost Rs/hr

        1

        175(MW)

        76.33

        37.21

        25.30

        38.5

        2.35

        1.41 x103

        2

        190(MW)

        83.31

        40

        27.65

        41.8

        2.8

        1.57 x103

        3

        220(MW)

        93.37

        45.73

        32.29

        48.4

        38

        1.73 x103

        4

        280(MW)

        123.86

        57.39

        41.40

        61.60

        6.26

        2.06 x103

        5

        320(MW)

        145.12

        65.35

        45.52

        70.4

        8.22

        2.29 x103

        6

        360(MW)

        164.59

        73.47

        47.34

        79.2

        10.45

        2.53 x103

        Genetic Algorithm

        Time (hr)

        PD (MW)

        P1 (MW)

        P2 (MW)

        P3 (MW)

        PH (MW)

        PL (MW)

        Fuel cost Rs/hr

        1

        175(MW)

        74.66

        39.05

        25.11

        38.5

        2.33

        1.41×103

        2

        190(MW)

        83.14

        40.52

        27.29

        41.8

        2.76

        1.57 x103

        3

        220(MW)

        96.82

        46.50

        32.04

        48.4

        3.77

        1.74 x103

        4

        280(MW)

        127.12

        56.67

        40.92

        61.60

        6.2

        2.06 x103

        5

        320(MW)

        144.23

        67.75

        45.52

        70.4

        7.91

        2.29 x103

        6

        360(MW)

        162.23

        76.46

        51.89

        79.2

        10.18

        2.53 x103

        7

        390(MW)

        183.94

        76.30

        56.05

        85.8

        12.91

        2.72 x103

        The total fuel cost is 6.2389e+004 Rs/Hr.

        fuel cost curve of 24 hours

        4000

        FUEL COST Rs/Hr.

        FUEL COST Rs/Hr.

        3000

        2000

        1000

        0

        0 5 10 15 20 25

        time

        Simulated Annealing

        Total fuel cost is 6.3352e+004 Rs/Hr.

        fuel cost curve of 24 hour

        5000

        fuel cost

        fuel cost

        4000

        3000

        2000

        1000

        0 5 10 15 20 25

        time

        Case Study 2: Six Unit System

        S.NO

        A

        B

        C

        Pmin

        P max

        1

        0.15247

        38.53973

        756.79886

        10

        125

        2

        0.10587

        46.15916

        451.32513

        10

        150

        3

        0.02803

        40.3965

        1049.9977

        35

        225

        4

        0.03546

        38.30553

        1243.5311

        35

        210

        5

        0.02111

        36.32782

        1658.5596

        130

        325

        6

        0.01799

        38.27041

        1356.6592

        125

        315

        Table: 7 Transmission loss (B-coefficients) six bus system

        1.4 x 10-4

        0.17 x10-4

        0.15 x10-4

        0.19 x10-4

        0.26 x10-4

        0.22 x10-4

        0.17 x10-4

        0.6 x10-4

        0.13 x10-4

        0.16 x10-4

        0.15 x10-4

        0.2 x10-4

        0.15 x10-4

        0.13 x10-4

        0.65 x10-4

        0.17 x10-4

        0.24 x10-4

        0.19 x10-4

        0.19 x10-4

        0.16 x10-4

        0.17 x10-4

        0.71 x10-4

        0.30 x10-4

        0.25 x10-4

        0.26 x10-4

        0.15 x10-4

        0.24 x10-4

        0.30 x10-4

        0.69 x10-4

        0.32 x10-4

        0.22 x10-4

        0.20 x10-4

        0.19 x10-4

        0.25 x10-4

        0.32 x10-4

        0.85 x10-4

        Time (hr)

        PD (MW)

        P1 (MW)

        P2 (MW)

        P3 (MW)

        P4 (MW)

        P5 (MW)

        P6 (MW)

        PH (MW)

        PL (MW)

        Fuel

        104 rs/hr

        1

        475

        13.01883

        10

        38.46605

        56.42657

        133.1914

        125

        104.5

        5.6029

        2.2128

        2

        490

        13.72726

        10

        42.19526

        59.34304

        137.8494

        125

        107.8

        5.9149

        1.2515

        3

        520

        15.14728

        10

        49.66613

        65.1853

        147.1738

        125

        114.4

        6.5725

        1.3465

        4

        580

        17.43105

        10

        61.71297

        74.53497

        161.9362

        134.8889

        127.6

        8.1040

        1.5411

        5

        620

        18.77937

        10

        68.83644

        80.03385

        170.5482

        144.6527

        136.4

        9.2505

        1.6743

        6

        660

        20.13339

        10

        75.98542

        85.55064

        179.1776

        154.4299

        145.2

        10.4769

        1.8104

        Time (hr)

        PD (MW)

        P1 (MW)

        P2 (MW)

        P3 (MW)

        P4 (MW)

        P5 (MW)

        P6 (MW)

        PH (MW)

        PL (MW)

        Fuel

        104 rs/hr

        1

        475

        13.01883

        10

        38.46605

        56.42657

        133.1914

        125

        104.5

        5.6029

        2.2128

        2

        490

        13.72726

        10

        42.19526

        59.34304

        137.8494

        125

        107.8

        5.9149

        1.2515

        3

        520

        15.14728

        10

        49.66613

        65.1853

        147.1738

        125

        114.4

        6.5725

        1.3465

        4

        580

        17.43105

        10

        61.71297

        74.53497

        161.9362

        134.8889

        127.6

        8.1040

        1.5411

        5

        620

        18.77937

        10

        68.83644

        80.03385

        170.5482

        144.6527

        136.4

        9.2505

        1.6743

        6

        660

        20.13339

        10

        75.98542

        85.55064

        179.1776

        154.4299

        145.2

        10.4769

        1.8104

        1. Lagrange Method

      Table: 8 six unit system result by Lagrange multiplier TOTAL FUEL COST =7.6570×105

      4

      4

      5 x 10

      FUEL COST CURVE OF SIX BUS SYSTEM

      FUEL COST

      FUEL COST

      4

      3

      2

      1

      0

      0 5 10 15 20 25

      TIME

      (A) Genetic Algorithm

      Table: 9 six unit system result for genetic algorithm

      Time (hr)

      PD (MW)

      P1 (MW)

      P2 (MW)

      P3 (MW)

      P4 (MW)

      P5 (MW)

      P6 (MW)

      PH (MW)

      PL (MW)

      Fuel rs/hr

      1

      475

      10

      11.45464

      38.83362

      38.99379

      151.4424

      133.9001

      104.5

      6.086173

      30089.87

      2

      490

      15.52055

      16.72736

      53.41049

      44.0457

      132.336

      125.9075

      107.8

      5.747647

      22227.95

      3

      520

      17.18647

      11.28117

      44.92188

      79.00682

      133.5426

      126.1406

      114.4

      6.479622

      23218.96

      4

      580

      11.17607

      12.86876

      88.90236

      72.65718

      142.0984

      132.4732

      127.6

      7.775892

      25335.33

      5

      620

      25.94088

      10.46391

      88.39069

      75.07086

      132.6921

      159.9954

      136.4

      8.953903

      26754.69

      6

      660

      21.41168

      12.71691

      89.26969

      81.08579

      158.7875

      161.7658

      145.2

      10.23743

      28147.11

      Total fuel cost =7.31×105 Rs/Hr

      4

      4

      4.5 x 10

      fuel cost curve of 24 hour (GA)

      4

      fuel cost

      fuel cost

      3.5

      3

      2.5

      2

      0 5 10 15 20 25

      1. time

Simulated Annealing

Table: 10 six unit system result for simulated annealing

Time (hr)

PD (MW)

P1 (MW)

P2 (MW)

P3 (MW)

P4 (MW)

P5 (MW)

P6 (MW)

PH (MW)

PL (MW)

Fuel rs/hr

1

475

13.02373

10.00004

38.46789

56.41934

133.1918

125.0001

104.5

5.602829

21664.3

2

490

13.72762

10.00002

42.20105

59.34261

137.8434

125.0001

107.8

5.914838

22173.13

3

520

15.1432

10.00001

49.66808

65.18661

147.1745

125.0001

114.4

6.572494

23199.57

4

580

17.42948

10

61.71116

74.53743

161.9387

134.8873

127.6

8.104057

25284.96

5

620

18.77888

10.00006

68.83556

80.03188

170.5451

144.6591

136.4

9.250587

26695.04

6

660

20.13047

10.00002

75.98726

85.54723

179.1745

154.4376

145.2

10.477

28120.77

Total fuel cost=7.3722×105

4

4

4.5 x 10

FUEL COST CURVE SIX UNIT SYSTEM (SIMULATED ANNEALING)

FUEL COST RS/HR

FUEL COST RS/HR

4

3.5

3

2.5

2

0 5 10 15 20 25

TIME

IV.SIMULATION RESULTS

fuel cost comparison for three unit system

Method

Fuel cost

Lagrange method

6.4557×104 Rs/Hr.

Simulated annealing

6.3352 x104 Rs/Hr

Genetic algorithm

6.2389 x104 Rs/Hr.

Fuel cost comparison for six unit system

Method

Fuel cost

Lagrange method

7.6570 x105Rs/Hr.

Simulated annealing

7.37 x105Rs/Hr

Genetic algorithm

7.31 x105 Rs/Hr.

CONCLUSION

In order to optimize the optimal hydro thermal generation scheduling carried out by lagrange simulated annealing and GA was employed to solve the problem while considering the constrains. These problem has been verified on the three different cases , three unit system six unit system and 15 unit system. The comparison of results for the test cases of three unit and six unit system and 15 unit system clearly shows that the GA method is more capable of obtaining higher quality solution. For the same power demand the fuel cost is minimized by employing GA method

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