# Optimal Short Term Hydrothermal Scheduling

Text Only Version

#### 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=0.006P1Â²+5.506P1+264.634 Rs/hr F2=0.016P2Â²+5.2P2+154.2 Rs/hr F3=0.005P3Â²+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

0.17 x10-4

 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 0.24 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

REFRENCES

1. Nagrath I.J and D.P Kothari, Power system Engineering, Tata McGraw-Hill New Delhi 1994.

2. Electrical energy systems theory- An Introduction ,McGraw- Hill, 1971

3. Power Dispatch Using Evolutionary Algorithm, Journal ofElectrical Engineering, vol. 57, no. 4, pp. 211-217, 2006.

4. I. G. Damousis, A. G. Bakirtzis, and P. S. Dokopoulos, Network- constrained economic dispatch using real-coded genetic algorithm, IEEE Trans. Power Syst., vol. 18, no. 1, pp. 198205,

Feb. 2003

5. T. Adhinarayanan and M. Sydulu, A directional search genetic algorithm to the economic dispatch problem with prohibited operating zones, in Proc. IEEE/PES Transmission and Distribution Conf. Expo.,Chicago, IL, Apr. 2124, 2008, pp. 15

6. B. K. Panigrahi, V. R. Pandi, and S. Das, Adaptive particle swarm optimization approach for static and dynamic economic load dispatch, Energy Convers. Manage., vol. 49, no. 6, pp. 14071415, 2008

7. C.-L. Chiang, Genetic-based algorithm for power economic load dispatch,IET Gen., Transm., Distrib., vol. 1, no. 2, pp. 261269, 2007

8. M. Basu, An interactive fuzzy satisfying-based simulated annealing techniques for economic emission load dispatch with nonsmooth fuel cost and emission level functions, Electric Power Components and Systems, vol.32, no.2, pp.163-173, 2004

9. S.Khamsawang, S.Pothiya and C.Booseng, Distributed Tabu Search for solving the economic dispatch problem,IEEE Trans. Power System., vol. 20, pp. 484487, 2004.

10. M. Basu, Particle Swarm Optimization based goalattainment method for dynamic economic emission dispatch, Electric Power Components and Systems, vol.34, pp.1015-1025, 2006.

11. S.Muralidharan, K.Srikrishna and S. Subramanian, Emission constrained economic dispatch- A new recursive approach, Electric Power Components and Systems, vol.34, no.3, pp.343- 353, 2006.

12. Whei-Min Lin, Fu-Sheng Cheng and Min-Tong Tsay, An Improved Tabu Search for Economic Dispatch with Multiple Minima, IEEE Trans. Power System. vol. 17, No.1, pp. 108- 112,

Feb 2002

13. A. Immanuel Selvakumar and K.Thanushkodi, A new particle swarm optimization solution to nonconvex economic dispatch problems,IEEE Transactions on Power Systems, Vol. 22, No. 1, pp.42-51, Feb. 2007

14. C.M. Huang and Y.C. Huang, A novelapproach to real-time economic emission power