Optimal Short Term Hydrothermal Scheduling

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

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

      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

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

Fuel cost comparison for six unit system

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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