Short Term Hydrothermal Scheduling using Evolutionary Programming

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Short Term Hydrothermal Scheduling usingEvolutionary Programming

Atul Kumar SharmaAsst. Manager (Engg. Wing)Airports Authority of India

Chandigarh, [email protected]

Rachna TyagiAssistant Professor

Electrical Engineering DepartmentAITM, Varanasi, India

[email protected]

S. P. Singh, Student Member, IEEE Electrical Engineering Department

IIT (BHU) VaranasiVaranasi, India

spsingh.rs.eee@iitbhu. ac.in

Abstract -- In this paper, Evolutionary Programming methodis used for short term hydrothermal scheduling which minimizethe total fuel cost while satisfying the constraints. This paperdeveloped and studies the performance of evolutionary programsin solving hydrothermal scheduling problem. The effectiveness ofthe developed program is tested for the system having one hydroand one thermal unit for 24 hour load demand. Numerical resultsshow that highly near-optimal solutions can be obtained byEvolutionary Programming.

Index Terms-- Hydro plant, Thermal plant, Evolutionary Programming, hydrothermal scheduling

I. I NTRODUCTION A modern power system consists of a large number of

thermal and hydel plants connected at various load centresthrough a transmission network. An important objective in theoperation of such a power system is to generate and transmit

power to meet the system load demand at minimum fuel cost byan optimal mix of various types of plants. The study of the

problem of optimum scheduling of power generation at various plants in a power system is of paramount importance,

particularly where the hydel sources are scares and high cost ofthermal generation has to be relied upon to meet the powerdemand. Cost optimization of hydro stations can be achieved byassuming the water heads constants and converting theincremental water (i.e. fuel) rate characteristics in to incrementalfuel cost curves by multiplying it with cost of water per cubicmeter and applying the conventional technique of minimisingthe cost function. Hydrothermal scheduling is required in orderto find the optimum allocation of hydro energy so that theannual operating cost of a mixed hydrothermal system isminimized.

Over the last decade the hydrothermal scheduling problemhas been the subject of considerable discussion in the powerliterature [1-3]. The available methods differ in the system

modelling assumptions and the solution. The annualhydrothermal scheduling problem involves the minimization ofthe annual operating cost of a power system subject to theseveral equality and inequality constraints imposed by the

physical laws governing the system and by the equipmentratings.

Different methods have been proposed for the solution ofthese problems in the past. Various methods, pontryaginsmaximum principle [4], general mathematical programming [5]and the dynamic programming [6] have been used to solve the

problem in different formulations. Methods based on Lagrangian

multiplier and gradient search techniques [7] for finding themost economical hydrothermal generation schedule under

practical constraints have been well documented. Mixed integer programming [8] methods have been widely used to solve suchscheduling problems in different formulations. Kirchmayer [9]utilized calculus of variation for short range scheduling problemand proposed the well known coordination equations. Among all

the methods, dynamic programming seems to be a popularapproach because it is usually satisfactory to find the economicdispatch at discrete load steps rather than at continuous loadlevels. In this respect stochastic search algorithms like simulatedannealing (SA) [10], genetic algorithm (GA) [11], evolutionarystrategy (ES) [12] and evolutionary programming (EP) [13-14]may prove to be very efficient in solving highly nonlinear HS

problems since they do not place any restriction on the shape ofthe cost curves and other non-linearities in model representation.Although these heuristic methods do not always guarantee theglobally optimal solution, they will provide a reasonablesolution (suboptimal near globally optimal) in a short CPU time.An Evolutionary programming method is used in this paperwhich minimizes the cost of short term hydrothermal

scheduling.The paper is organized as follows: Section I summarized theoverview of the problem, brief literature review, Section IIhighlights the Hydrothermal Scheduling and also gives theoverview of various hydro plants and classification of problemformulation and various solution approaches. The problemformulation using EP is discussed and the algorithm for the EP

based HS problem is presented in section III. Finally test resultsare given in section IV and section V concludes the paper.

II. SCHEDULING E NERGY Suppose, as in Fig. 1, we have two sources of electrical

energy to supply a load, one hydro and another steam. Thehydro plant can supply the load by itself for a limited time.

Fig. 1 Two-unit hydrothermal systems

International Journal of Inventions in Reasearch, Engineering Science and Technology (IJIREST),Vol.1,No.1,April 2014ISSN(Print):2348-7399 ISSN(Online):2348-8077

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For any time period j ,

(1)

= 1 However, the energy available from the hydro plant isinsufficient to meet the load

(2) = number of hours in period j

= = (3) Use here the entire amount of energy from the hydro plant insuch a way that the cost of running the steam plant isminimized. The steam-plant energy required is

= (4) (Load energy) - (Hydro energy) = (Steam energy)

It is not require the steam unit to run for the entire interval ofTmax hours. Resulting optimal hydro thermal schedule shows ifthe thermal and hydro plants are operated with each other thenit results in economy. Steam plants and hydro plants, both areused to supply base load. During peak load it is economical touse hydro plants. Hence the thermal plant is preferred as a baseload plants whereas the hydroelectric plant is run as a peak load

plant

A. Thermal ModelThe objective function is to minimize the total operating

cost (C) represented by the fuel cost of thermal generation overthe optimization interval (T).

= ( ) (5) where the problem is to schedule the power generation of allunits over t k time sub -intervals in order to minimize the fuelcost which is given as :

( ) = + + (6) = 1 , = 1..

Where a i , b i and ci are cost coefficients of the it h generating unit.

B. Hydro Model

In hydro system, there is no fuel cost incurred in theoperation of hydro units. According to Glimn-krichmayermodel, discharge is a function of power output and the head.For large capacity reservoir it is practical to assume that theeffective head is constant over the optimization interval. Thusq jk is the rate of discharge from the j th unit in the interval k andis represented by the quadratic equation:

= , + , + (7) Where x j , y j and z j discharge coefficients of the hydro units.

C. Constraints Load Demand Equality Constraint

= P + P (8) where Pdk is the load demand during the k

th sub- interval andP lossk are the transmission losses during the k

th interval.Minimum and Maximum Power Generation Limits from ViewPoint of Economy and Capacity of Generating Units

P P P (9) where P ik Power output of the generating units in MW duringthe k th interval, P max is the maximum power of a generating unitin MW and P min is the minimum power of a generating unit inMW.

D. Transmission LossesThe transmission losses during k th interval are given by theKrons loss formula in terms of B- coefficients.

= + + (10) The fixed head hydro thermal problem can be definedconsidering the optimization interval to meet the load demandin each interval. Each hydro plant is constrained by the amountof water available for draw down in the interval.

III. EP BASED SHORT TERM HYDROTHERMAL SCHEDULING Evolutionary programming is based on Finite State

Machine (FSM) model in its early stage, which is proposed byL. J . Fogel in 1960's when he studied the artificial intelligence.In 1990's , the evolutionary programming was developed by D.B. Fogel and was made to solve the optimal problems in realspace. The evolutionary programming has been become anoptimal tool and was used in many practical problems. EP is aartificial intelligence method in which an optimisationalgorithm is the main engine for the process of three steps,namely, natural selection, mutation and competition. Accordingto the problem, each step could be modified and configured inorder to achieve the optimum result. Each possible solution tothe problem is called an individual. In order to use EP, themathematical model should be capable of dealing with the datatype and structure of individuals.

A. Elements of Evolutionary Programming

EP consists of a number of elements that are required forsuccessful implementation. Evolutionary Programming requirescertain steps, components, procedures or operators that must bespecified. The main elements are:

1. Representation2. fitness function3. Population4. Parent selection methodology5. mutation6. Survivor selection mechanism (replacement)


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7. Initialization8. Termination condition

To obtain a running algorithm the initialization procedure and atermination condition must be also defined.

B. Basic Algorithm of Evolutionary Programming1. The initial population (parent) is determined with the

help of random number. The random number is anormally distributed random number with in a

particular range.2. Calculate the fitness of parents.3. Make offspring with the help of parents. Calculate

fitness of each offspring.4. Combine all parent and offspring in a mating pool.5. Start a game for competition and selection.6. Select a competitor at random from the trial solutions.

Let it is Q r. where

r = (2N pu2 + 1), N p = Number of individuals in a population. u 1, u2 = uniform random numbers rangingover [0,1].

7. Assigns wins for each competitor as-8. Wn = 1, if u 1 < (fitness of Q r )./( fitness of Q r + fitness

of Q i).Wn = 0, otherwise.

9. Add all the wins for all the competitors.10. Arrange in descending order all the competitors as per

their wins11. Select the top N p among the competitor and they will

be new parent.

C.

Algorithm of Evolutionary Programming for the Test Case1. The problem variables to be determined are

represented as a j-dimensional trial vector, where eachvector is an individual of the population to be evolved.

2. An initial population of parent vectors Q k for k =1,2....N p is selected at random from a feasible range ineach dimension . The distribution of these initial

parent vectors is uniform.3. An offspring is generated by from each parent with

adaption of strategy parameter based on scaled cost.4. Fitness function is evaluated for each individual of

both parent and child populations.5. A competitor is chosen randomly from the combined

population of 2Np trial solutions (N p parents and N p off springs) and stochastic competition is performed based on the value of fitness function whereeach individual in the competing pool competesagainst other members for survival.

6. After the competition is over, the 2N p trial solutions inthe competing pool are sorted according to their scoresfrom the highest to the lowest. Thereafter the first, trial

solutions are selected as the new parent vectors for thenext generation.

7. If current generation is greater than or equal to themaximum generation, print the result and stop;

otherwise repeat steps 3 to 6.IV. R ESULT AND DISCUSSION

In this section, the results of Short term hydro thermalscheduling problem after the implementation of EP arediscussed. Test system is taken from the Appendix A. Thealgorithms are implemented in MATLAB to solve the problem.The main objective is to minimize the cost of generation ofthermal plants.Developed program gives the Water discharge, Thermal andHydro generations and total operating cost. The results areshown in the Table I and Table II.

TABLE I. Thermal and Hydro generations for the given loaddemand

KUsing EP Using Classical Method

PDK (MW)

P1K (MW)

P2K (MW)

PDK (MW)

P1K (MW)

P2K (MW)

1 455.0 226.3438 228.7348 455.0 225.3438 229.23482 425.0 200.7639 224.1628 425.0 199.9639 224.96283 415.0 189.8766 225.1323 415.0 188.8766 226.13234 407.0 183.5105 223.4971 407.0 182.3105 224.49715 400.0 176.0357 223.9646 400.0 175.1357 224.86466 420.0 195.4542 224.5523 420.0 193.9542 226.02307 487.0 253.6326 233.3688 487.0 251.5096 235.49188 604.0 361.6458 242.3672 604.0 359.6658 244.34729 665.0 418.9194 246.0860 665.0 416.4634 249.0060

10 675.0 428.4532 246.5500 675.0 427.1332 247.8700

11 695.0 445.2664 249.7380 695.0 442.1664 252.838012 705.0 456.2759 248.7291 705.0 453.3759 251.629113 580.0 340.5407 239.4549 580.0 338.1907 241.764914 605.0 363.5747 241.6224 605.0 359.4747 245.727415 616.0 374.3815 241.6224 616.0 371.8815 241.122416 653.0 408.3701 244.6300 653.0 406.2301 246.770017 721.0 471.6690 249.3368 721.0 468.5460 252.459818 740.0 490.8071 249.1931 740.0 488.5721 251.428119 700.0 453.3001 246.7033 700.0 450.8001 249.203320 678.0 431.7400 246.2615 678.0 430.5055 244.356021 630.0 385.1530 244.8517 630.0 382.8530 247.151722 585.0 344.6537 240.3400 585.0 341.2537 243.740023 540.0 306.4835 233.9242 540.0 306.4835 236.7242

24 503.0 271.5632 231.4425 503.0 269.2176 233.7861

For the simplification of problem the transmission losses areneglected. With the application of Classical Method on the testcase, thermal and hydro generations and hourly cost anddischarge for the given load demand are obtained.Total discharge for 24 hour load demand = 72.7743 Mm 3 Total operating cost for 24 hour load demand = Rs.92,791.2719.


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TABLE II . Hourly cost and discharge for the given loaddemand

KUsing EP Using Classical Method

PDK (MW)

q 1K(Mm 3/h)

FK(Rs/h)

PDK (MW)

q 1K(Mm 3/h)

FK(Rs/h)

1 455.0 2.6792 2645.8103 455.0 2.9543 2639.45512 425.0 2.6325 2379.2227 425.0 2.9106 2374.16443 415.0 2.9123 2269.4363 415.0 2.9225 2259.07624 407.0 2.8922 2203.5500 407.0 2.9079 2191.14975 400.0 2.9001 2126.3971 400.0 2.9096 2117.12256 420.0 2.9064 2327.2939 420.0 2.9115 2311.72197 487.0 2.9973 2938.1747 487.0 2.9754 2915.64608 604.0 3.0935 4108.0678 604.0 3.0729 4086.20449 665.0 3.1344 4747.2472 665.0 3.1145 4697.0492

10 675.0 3.1395 4854.9135 675.0 3.1542 4839.985011 695.0 3.1756 5045.6690 695.0 3.2100 5010.413012 705.0 3.1637 5171.1880 705.0 3.1963 5138.078313 580.0 3.0620 3875.8261 580.0 3.0870 3850.0764

14 605.0 3.0834 4129.3820 605.0 3.1303 4084.095015 616.0 3.0854 4249.0702 616.0 3.0800 4221.340716 653.0 3.1183 4628.5345 653.0 3.1419 4604.506917 721.0 3.1705 5347.4934 721.0 3.2057 5311.648918 740.0 3.1689 5568.0081 740.0 3.1941 5542.180619 700.0 3.1412 5137.2133 700.0 3.1178 4913.748320 678.0 3.1363 4892.1157 678.0 3.1018 4878.137721 630.0 3.1207 4368.8302 630.0 3.0926 4343.219522 585.0 3.0716 3920.9467 585.0 3.0381 3883.643123 540.0 2.9989 3504.7993 540.0 2.9719 3444.501024 503.0 2.9772 3127.8755 503.0 2.9772 3104.1081

With the application of EP on the test case, thermal and hydrogenerations and hourly cost and discharge for the given loaddemand are obtained.Total discharge for 24 hour load demand = 72.7603 Mm 3

Total operating cost for 24 hour load demand = Rs. 93,567.0655Thus it can be concluded that Evolutionary programmingtechnique provides optimum results as close to the Classicalmethod. Although cost obtained by Classical method is less ascompare to that obtained by Evolutionary programmingtechnique, it is better to use Evolutionary programming becauseClassical method can not be applied to the hydrothermalscheduling problem having prohibited zone constraints.Classical method has the major disadvantage of drastic growthof computational requirements with increasing system size.While implementing EP there is no need of initial guess of

power and water discharge. Hence it is better to use EP even itgives total operating cost more than that given by Classicalmethod.

V. CONCLUSION An evolutionary programming based approach has been

proposed and demonstrated to solve the short termhydrothermal scheduling problem. Numerical results show thathighly near-optimal solutions can be obtained by EP. Theeffectiveness of the developed program is tested for the system

having one hydro and one thermal unit for 24 hour loaddemand. The EP-based algorithm is faster in searching theoptimal solution due to implicit parallelism employed in EP.Since the encoding and decoding schemes are not needed in the

proposed EP approach, a lot of computer memory andcomputing time can thus be saved. As only the objectivefunction is relied on to guide the search towards the optimalsolution in the EP-based algorithm, the generation models ofeither hydro or thermal units can be of nonlinear and nonsmooth functions with prohibited areas. So it is clear that withthe help of EP-based algorithm it is possible to find a morenearly optimal solution to the existing hydrothermal scheduling

problem.

VI. R EFERENCES 1. Quintana V.H., Chikhani A.Y., "A Stochastic Model for Mid-

term Operation Planning of Hydro-Thermal Systems withRandom Reservoir Inflows", IEEE Trans. on Power Apparatusand Systems, Vol.PAS-100, No.3, March 1981.

2. Davist C.E., Pronovost R., "The Stochastic DynamicProgramming Procedures for Long Term ReservoirManagement", IEEE Trans. on Power Apparatus and Systems,Vol.PAS-91, No.6, Nov/Dec 1972.

3. Turgeon A., "Optimal Operation of Multireservoir Power Systemswith Stochastic Inflows", Water Resources Research, Vo1.16,

No.2, April 1980.4. Dahlin E.B., D.N.C. Shen, Optimal Solution to the Hydro-Steam

Dispatch Problem for Certain Practical systems, IEEE Trans. onPAS, Vol. PAS-85, May 1966, pp.437-458

5. Habibollahzadeh H., Frances D., Sui U. A New GenerationScheduling Problem At Ontario Hydro, IEEE Trans. on powersystems, vol. 5, No.1, Feb, 1990, pp 65-73

6. Amado S.M., Rebeiro C.C., Short-Term Generation Schedulingof Hydraulic Multi Reservoir Multi Area InterconnectedSystems, IEEE Trans. on power systems, vol. PWRS-2,

No.3,Aug.1987, pp.758-763.7. Soares S., Lyra C., Tavares H., Optimal Generation Scheduling

of Hydrothermal Power Systems, IEEE Trans. on PAS, Vol.PAS -99,No.3,May/June 1980, pp.1107-1175.

8. Tang J., Luh P.B., Chang T.S., Mixed Co-Ordination Method for Nonlinear Programming Problems with Separable Structures,International Journal of control, Vol.50, No. 4, 1989, pp.1461-1486.

9. Kirchmayer L.K., Economic Control of InterconnectedSystems, Wiley, New York (1959).

10. Wong K.P. and Wong Y.W., Short-Term HydrothermalScheduling, Part-I: Simulated Annealing Approach. IEE Proc.Part-C 141 5 (1994), pp. 497501.

11. Wu Y.-G., Ho C.-Y. and Wang, A Diploid Genetic Approach toShort-Term Scheduling Of Hydrothermal System . IEEE Trans.

Power Syst. 15 4 (2000), pp. 12681274.12. Bck T. and Schwefel H.-P., An Overview of EvolutionaryAlgorithms for Parameter Optimization. Evol. Comput. 1 1(1993), pp. 123

13. Fogel D.B., An Introduction To Simulated EvolutionaryOptimization. IEEE Trans. Neural Netw. 5 1 (1994), pp. 314.

14. Sinha N., Chakrabarti R. and Chattopadhyay P.K., EvolutionaryProgramming Techniques for Economic Load Dispatch. IEEETrans. Evol. Comput. 7 1 (2003), pp. 8394.


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VII. APPENDIX AThe performance is evaluated for the following case:

The test system consists of one thermal and one hydrogenerating station as shown in figure. The operating cost isgiven by-

( ) = + +The rate of discharge of hydro generating station is given by- = + +

The coefficients of fuel cost are given in table A.1 (a). Thedischarge coefficients of hydro plants are given in table A.1(b). Maximum and minimum power limits are given in tableA.1(c). The load demands for 24-hour intervals are shown intable A.1 (d).

Fig. A.1 Test system

Unit no. a 1 b1 c1 1 0.001991 9.606 373.7

(a)

Unit no. x 1 y1 z1 2 2.19427*10^- 4 2.5709*10^ - 4 1.742333

(b)

Limits Max MinP1 500 150P2 300 100

(c)

KLoad demand(PDK )(MW)

KLoad demand(P DK )(MW)

1 455.0 13 580.02 425.0 14 605.03 415.0 15 616.04 407.0 16 653.05 400.0 17 721.06 420.0 18 740.07 487.0 19 700.08 604.0 20 678.09 665.0 21 630.0

10 675.0 22 585.011 695.0 23 540.012 705.0 24 503.0

(d)TABLE A.1. Specifications of test system

(a) Cost coefficients(b) Discharge coefficients(c) Power generation limits(d) Load demand Energy Syst. Vol. 42, pp. 71-77, 2012.

VIII. B IOGRAPHIES

Atul Kumar Sharma has done his B.Tech from Dr. KNMIET,Modinagar in Electrical Engineering in 2006 and M.E. fromThapar University, Patiala in 2009 with specialization in PowerSystem & electric drives. In 2009 he joined Airports Authority

of India & now working as Asstt. Manager in EngineeringWing. His interest towards study & research is still alive.

Rachna Tyagi was born in Meerut, India. She received theB.Tech. degree in Electrical Engineering from the UttarPradesh Technical University (UPTU), Lucknow,, India, andthe M.Tech. degree from Indian Institute of Technology (IIT),Delhi, India, in 2007, 2013 respectively. Her research interestsinclude Solar Energy and its applications in solar distillation,controlled environment greenhouse, crop production anddrying, renewable energy resources, energy analysis of allsystems, hybrid PV/thermal systems.

Satyendra Pratap Singh was born in Varanasi, India, in 1985.

He received the B.Tech. degree in electrical engineering fromthe Uttar Pradesh Technical University (UPTU) Lucknow,India, in 2007, and the M.E. degree in electrical engineeringwith specialization in power system and electric drives fromthe Thapar University, Patiala, India, in 2009. He is currently

pursuing the Ph.D. degree in electrical engineering withspecialization in power system from the Indian Institute ofTechnology (BHU) Varanasi, India. His research interestsinclude wide area measurement in power system, power systemoperation, applications of AI in power system optimization.


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Short Term Hydrothermal Scheduling using Evolutionary Programming

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