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International Journal of

Mathematical Modelling& ComputationsSummer 2014

Aims and Scope

International Journal of Mathematical Modelling & Computations (IJM2C) isan international scientific journal and a forum for the publication of high-qualitypeer-reviewed papers published by academic publications of Islamic Azad Uni-versity, Central Tehran Branch (IAUCTB), which is aimed to publish originaland high-quality research articles on all subareas of computational and applied

mathematics and related topics such as algorithms and complexity, approxima-tion theory, approximate reasoning, calculus of variations, chaotic systems, com-putational intelligence, computer simulation, cryptography, design of algorithms,discrete mathematics, dynamical systems, fuzzy logic, fuzzy set theory and fuzzysystems, graph theory, integral equations, intelligent systems, mathematical mod-elling, numerical analysis and computations, neural networks, nonlinear analysis,operation research, optimal control theory, optimization, ordinary and partial dif-ferential equations, special functions, soft computing and their applications to anyfield of science. Also, this journal will offer an international forum for all scien-tists and engineers engaged in research and development activities. It promotesthe application of mathematics to physical problems particularly in the area of

engineering. The journal invites original papers, review articles, technical reportsand short communications containing new insight into any aspect of applied sci-ences that are not published or not being considered for publication elsewhere. Thepublication of submitted manuscripts is subject to peer review and both generaland technical aspects of the submitted paper are reviewed before publication.

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International Journal of

Mathematical Modelling& ComputationsSummer 2014

Call for Papers

Dear Colleagues,

International Journal of Mathematical Modelling& Computations(IJM2C) is ded-icated to the publication of high quality and peer reviewed original research, tech-nical advances and contributions in all area of applied mathematics, mathematical

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Several analytical and numerical models have been developed to predict the

surface-groundwater interaction in stream-aquifer systems under varying hydrological

conditions [11-14, 18]. Estimations of water table fluctuations in unconfined aquifer with

surface infiltration have also been presented by several researchers [15, 17, 19-21].

The subsurface seepage flow in unconfined aquifers is usually formulated as a parabolic

nonlinear Boussinesq equation which does not admit analytical solution. Therefore, themethodology adopted in most of the aforesaid models is to linearize the Boussinesq

equation and develop analytical solution of the resulting equation. Although, theapproximate analytical solution obtained in these studies provide useful insight in the flow

process; however, the subsurface drainage over hillslope cannot be satisfactorily addressed

with their results. Another perceived limitation is that they do not account for the gradualrise in the stream water.

The role of sloping bedrock in evolution of the phreatic surface has been underlined in

numerous studies [8-10]. The results derived for horizontal aquifer may seriously

underestimate or overestimate the actual results if applied to the sloping bed geometry.

Approximation of groundwater flow over unconfined sloping beds based on extendedDupuit-Forchheimer assumption (streamlines are approximately parallel to the sloping

impervious bed) can be expressed in the form of an advection-diffusion equation,

popularly known as Boussinesq equation [1, 7]. Analytical solutions of linearizedBoussinesq equation under constant or time-varying boundary conditions have been

presented by a number of researchers [1- 6, 16]. It is worth mentioning that the efficiencyof the linearization must be examined by validating the model results with either field data

or by comparing the results with numerical solution of the nonlinear Boussinesq equation.

The present study is an attempt to quantify the groundwater-surface water interaction with

more realistic approach. The mathematical model considered here deals with transientgroundwater flow regime in a homogeneous unconfined aquifer of finite width overlying a

downward sloping impervious bed owing to seepage from stream of varying water leveland constant downward recharge. The aquifer is in contact with a constant piezometric

level at one end and a stream of time varying water level at another end. The stream isconsidered to penetrate full thickness of the aquifer. Furthermore, the aquifer is

replenished by a constant recharge. Efficiency of the linearization method is examined by

solving the nonlinear Boussinesq equation by a fully explicit predictor-corrector numericalscheme. The effect of bed slope, recharge rate and stream rise rate on the water table

fluctuation and flow mechanism is analyzed using a numerical example. The solution

presented here can be applied to asymptotic scenarios of sudden or very slow rise in thestream water by assigning an appropriate value to the stream rise rate parameter.

2.Problem Formulation and Analytical Solution

As shown in Fig.1, we consider an unconfined aquifer overlaying an impermeable sloping

bed with downward slopetan. The aquifer is in contact with a constant piezometriclevelat its left end and a stream at the right end. The water in the stream is graduallyrising from its initial level to a final level by a known exponential decayingfunction of time. Moreover, the aquifer is replenished vertically at a constant rate. If thevariation in the hydraulic conductivityand specific yield of the aquifer with spatial

coordinate is neglected, and the streamlines are considered to be nearly parallel to the

impermeable bed (extended Dupuit-Forchheimer approach) then the groundwater flow inthe aquifer can be characterized by the following nonlinear Boussinesq equation [10]

(1)

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where,is the height of the water table measured above the impermeable slopingbedin the vertical direction. andrespectively are the hydraulic conductivity and specificyield of the aquifer. is the constant recharge rate. is a unit step function defined asfollows: 0 01 0 (2)

Fig. 1.Schematic diagram of an unconfined aquifer

The initial and boundary conditions are:

, 0 (3) 0, (4)

, (5)whereis length of the aquifer.is a positive constant signifying the rate at which the waterin the stream rises from its initial value to a final level . Equation (1) is a secondorder nonlinear parabolic partial differential equation which cannot be solved by analytical

methods. However, an approximate analytical solution can be obtained by solving the

corresponding linearized equation. In the present work, we adopt the linearization methodas suggested by Marino [16]. Firstly, rewrite Equation (1) as

(6)where is the mean saturated depth in the aquifer. The value of is successivelyapproximated using an iterative formula

/2, where

is the initial water

table height, and is the varying water table height at time at the end of which isapproximated. Equation (6) is further simplified using the following dimensionlessvariables and substitutions

hL

N

Stream

Initial Level

L

Initial water head

h0

x

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, , (7)

Therefore, Equation (6) becomes

(8)

where

0 if 01 if 0 (9)Now, define the following parameters

Ltan 2h , N LNKhh h cos , SL

Khcos (10)

so that, equation (8) becomes

HX 2 HX N H (11)The initial and boundary conditions reduce to

HX, 0 X (12)HX 0, 0 (13)

HX 1, 1 e (14)Equation (11) along with conditions(12)(14) is solved by Laplacetransform. Define theLaplace transform of HX,as follows:

LHX, : s HX, s HX, ed (15)The Laplace transform of equation (11) yields

HX 2 HXNs H X (16)The general solution of equation (16) can be found using elementary methods. Moreover,the arbitrary constants contained in the general solution are obtained by using the Laplace

transform of Equations (13)(14) in it. We get

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, 2 1

2

2 (17)

Inverse Laplace transform of equation (17) is obtained from calculus of residue. Define

, , 12 lim , (18)wherecis an arbitrary positive number. The integration in equation (18) is performed alonga lines = cin the complex plane wheres = x + i y. The real number cis chosen so thats =

c lies to the right of all singularities, but is otherwise arbitrary. The inverse Laplace

transform of equation (17) yields

, 1 22 1

11

2

11

(19)

Equation (19) provides analytical expression for the water head distribution in the

downward sloping aquifer under conditions mentioned in Equations (3)(5). One can

obtain the corresponding results for an upward sloping by replacing by and for ahorizontal bed by setting 0. Furthermore, equation (19) can also be used to predict thewater head profiles when the rise in the stream is extremely rapid; similar to the case of

flood like situation, by letting . We obtain

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, 1 22 1

1

1

2 11

(20)

3.Determination of Flow Rate and Steady State Profile

The flow rate, in the aquifer is defined as

, tan

(21)

andat the stream-aquifer interface, the flow rate is

tan (22)Invoking Equation (20) in Equation (22), the expressions for flow rate at the

stream-aquifer interface are obtained as follows:

1 22

1

1

1

2 1

1 e tanh tan

(23)

It is worth noting that despite continuous recharge, the profiles of water head attain asteady state value for large value of time. The expressions for steady-state water head andflow rate at stream-aquifer interface can be obtained by setting t in Equations (20) and

(23), yielding

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/

22

/

/

1/ 2/ 1/

(24)

1

22 1

2

(25)

4.Numerical Solution of the Non-linear Equation

To assess the efficiency of linearization technique, the non-linear Boussinesq equation issolved numerically by using Mac Cormack scheme. For this, Equation (1) is written as

(26)where, cos / and sin2/2. Mac Cormack scheme is a predictorcorrector scheme in which the predicted value of his obtained by replacing the spatial and

temporal derivatives by forward difference, i.e.

, , ,, ,

,, , , , NSt(27)

The corrector is obtained by replacing the space derivative by backward differences,

whereas the time derivative is still approximated by forward difference

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, , , , , , , ,

, , N

St

(28)

The final value of , is given as an arithmetic mean of , and , , i.e., 12 , , , , , , , , , , NSt

(29)

The initial and boundary conditions are discretized as, (30)

, (31)

, (32)Numerical experiments reveal that the method is stable if

0.06 and 0.095. Discussion of Results

To demonstrate the applicability of the closed form solution given by Equations (19) and

(23), we consider an aquifer with = 150 m. Other hydrological parameters are: = 3.6m/h, = 0.34, = 5 m, = 2 m, = 4 mm/hr and = 0.054 per hr. Numericalexperiments were carried out, and it is found that the first 50 terms of the summation series

satisfactorily approximates the final value. The profiles of transient water head

obtained

from equation (19) for = 3 deg are plotted in Fig. 2 (continuous curves).

Fig. 2.Comparison of analytical and numerical solution for = 3 deg2

2.5

3

3.5

4

4.5

5

5.5

6

0 50 100 150

Waterheadh(m)

Distance x (m)

Analytical

--- Numerical 10h

20h

40h

80h

t=120h

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Numerical solution of the non-linear Boussinesq equation (1) using Mac Cormack schemefor the same data set is also presented in Fig. 2 (dotted curves). Close agreement between

the numerical and analytical solutions demonstrates the efficiency of the linearization

method adopted in this study. Comparison of water head height obtained from numericaland analytical solutions is presented in Table 1. It is observed that the analytical solution

slightly underestimates the actual results.

Table1. Comparison of analytical (anal.) andnumerical (num.) resultsx(m) t=20h

anal. num.

t=40h

anal. num.

t=80h

anal. num.

t=120h

anal. num.

20 4.8216 4.8466 4.9321 4.9492 5.0956 5.0964 5.175 5.177540 4.5558 4.6001 4.7919 4.8213 5.14 5.1403 5.3005 5.3046

60 4.2301 4.2853 4.6054 4.6380 5.1479 5.1462 5.3791 5.3838

80 3.879 3.9341 4.4116 4.4387 5.1336 5.1297 5.4103 5.4153

100 3.5611 3.6002 4.2742 4.2905 5.1081 5.1032 5.3891 5.3942120 3.3997 3.4185 4.2817 4.2846 5.0731 5.069 5.3036 5.3079

140 3.6527 3.6525 4.49 4.4845 5.0119 5.0108 5.1293 5.1312

Relative percentage difference (RPD) between numerical and analytical values of waterhead with the numerical solution is analyzed. It is observed that the RPD is maximum in

the middle part of the aquifer and negligible near the interfaces

0 and

. In

order to obtain a validity range of the analytical results, the range of RPD for different

values of sloping angle is presented in Table 2.Table2. Range of RPD between analytical and numerical solution for = 0.054 h-1 Range of RPD

5 0.0713 to 0.5232

3 0.1322 to 0.5455

0 0.1822 to 0.5692

3 0.2014 to 0.6227

5 0.2511 to 0.6532

Average distance between the analytical and numerical solutions is also calculated using

2 norm which is defined as follows:

1 1 / (33)whereand respectively denote the numerical and analytical solutions. The 2norm for 3 deg, 0.054h, 20, 40, 80 and 120 hrs is presented in Table 3.

Table3. Average distance between analytical and numerical solution

t(h) Average distance using

L2 Norm

20 0.01344

40 0.02107

80 0.01074

120 0.00528

Fig. 3 presents the comparison of transient water head profiles for and 5 deg. Astime progresses, the water table grows in the aquifer. Continuous recharge causes

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groundwater mound which finally stabilizes due to inflow-outflow equilibrium. Since,

steeper downhill slope allows more water to from the left end; such aquifers exhibit

relatively higher growth the phreatic surface. This phenomenon is evident from Fig. 3

wherein the water table profiles in an aquifer with 5 deg bed slope are higher than the

corresponding profiles of 3 deg bed slope.

Fig. 3.Transient profiles of water head for = 3 and 5 degConsidering a fixed pointxin the domain with distance ratiox/L= 0.75, we plot in Fig. 4

the water head at this point against time tfor various values of. It can be observed from

this figure that the rise rate of the stream water plays an important role in determining thetransient profiles of free surface as well as its stabilization level. A fast rising stream

reduces the outflow at stream-aquifer interface, leading to growth in the water table.

Fig.4.Variation in water head height atx = 75 m

Using equation (23), the flow rates at interfacex=Lis plotted in Fig. 5. The initial flowrate at the interfacex=Lare given by

tan (34)

2

2.5

3

3.5

4

4.5

5

5.5

6

0 50 100 150

Waterheadh(m)

Distance x (m)

10h

20h

40h

80h

t=120h

=3Deg

=5Deg

3.5

4

4.5

5

5.5

0 20 40 60 80 100

Waterheadh(m

)

Time t (hr)

Sudden rise

= 0.054 h-1

- = 0.05 h-1

- - - - = 0.012 h-1

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Clearly, varies with bed slope. At initial stages, the stream-aquifer interfaceexperiences flow along positive direction of x-axis. As time progresses, the outflow at reaches a minimum level and increases thereafter. A reasonably long aquifer withsmall bed slope experiences a temporary inflow at this interface. For large value of time,

the flow rate attains steady state value given by Equation (25). It is worth noting thatstabilization of flow rates leads to convergence of water table to a final value. Variation in

with recharge rate

and stream rise rate

are plotted in Fig. 6 and Fig. 7

respectively. It is clear from this figure that flow rate at stream-aquifer interface increaseswith recharge. Furthermore, stream rise rate plays an important role in determining the

inflow-outflow at the stream aquifer interface. It is established that a fast rising stream may

allow its water to enter into the aquifer (bank storage mechanism).

Fig. 5.Flow rate at the stream-aquifer interface for = 3 and 5 deg

6. Conclusions

This study focuses on three major issues: (i) derive analytical expressions for water head

and flow rate in an unconfined sloping aquifer due to continuous recharge and

stream-varying water level, (ii) examine the efficiency and validity range of thelinearization of method, and (ii) analyze the response of an aquifers to varying

hydrological parameters. The linearized Boussinesq equation characterizing the transient

groundwater flow is solved using Laplace transform technique, and the corresponding

nonlinear equation is solved numerically by a fully explicit predictor-corrector scheme.Numerical experiments carried out in this study indicate that the evolution and

stabilization of the phreatic surface and interaction of water between the stream andaquifer depend significantly on the bed slope and other aquifer parameters.

Fig. 6.Variation in flow rate at stream-aquifer interface withN

-0.5

0.5

1.5

2.5

3.5

0 50 100 150 200

FlowRates(m/h)

Time t (h)

= 5 deg

= 3 deg

-0.5

0.5

1.5

2.5

0 50 100 150

FlowRateqx

=L(

m/h)

Time t (h)

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Fig. 7.Variation in flow rate at stream-aquifer interface with

7. References

[1] Akylas, E., Koussis,A.D., Response of sloping unconfined aquifer to stage changes in adjacent stream. I.Theoretical analysis and derivation of system response functions. J. Hydrol,338(2007) 8595.

[2] Bansal, R.K., Das, S.K., The effect of bed slope on water head and flow rate at the interfaces between the streamand groundwater: analytical study. J HydrolEng,14(8) (2009) 832838.

[3] Bansal, R.K., Das,S.K. Water table fluctuations in a sloping aquifer: analytical expressions for water exchange

between stream and groundwater. J Porous Media ,13(4)(2010) 365374.

[4] Bansal, R.K. Das S.K.An analytical study of water table fluctuations in unconfined aquifers due to varying bedslopes and spatial location of the recharge basin. J HydrolEng, 15(2010) 909917.

[5] Bansal, R.K., Das S.K., Response of an unconfined sloping aquifer to constant recharge and seepage from the

stream of varying water level.Water Resour. Manag,25(2011) 893911.

[6] Bansal R.K., Groundwater fluctuations in sloping aquifers induced by time-varying replenishment and seepage

from a uniformly rising stream. Trans. Porous Med,94(2012) 817-826.

[7] Behzadi,S.S., Numerical solution of Boussinesq equation using modified Adomian decomposition and homotopyanalysis methods. Int. J. of Mathematical Modeling & Computations, 1(1)(2011) 4558.

[8] Boufadel, M.C. PeridierV., Exact analytical expressions for the piezometric profile and water exchange between

the stream and groundwater during and after a uniform rise of the stream level. Water Resour Res. Doi:10.1029/2001WR000780, 2002.

[9] Brutsaert, W., The unit response of groundwater outflow from a hillslope. Water Resour Res,30(10) (1994)

27592763.

[10]Chapman T.G., Modelling groundwater flow over sloping beds. Water Resour Res,16(6) (1980) 11141118.[11]Gill,M.A., Bank storage characteristic of a finite aquifer due to sudden rise and fall in the river level. J Hydro,

l76(1985) 133142.

[12] Hantush,M.S., MS Wells near streams with semipervious bed. J Geophys Res,3(1) (1965), 227234.

[13]Higgins,D.T., Unsteady drawdown in a two-dimensional water table aquifer. J Irrig Drain Div, Proc AmSocCivEng 106(IR3) (1980)237251.

[14] Latinopoulos,P., Periodic recharge of finite aquifers from rectangular areas. Adv Water Resour, 7(1984) 137140.[15] Latinopoulos,P., A boundary element approach for modeling groundwater movement.Adv Water Resour,9(1986)

171177.

[16] Marino M.A., Rise and decline of the water table induced by vertical recharge. J. Hydrol,23(1974) 289298.[17]Moench, A.F., Barlow P.M., Aquifer response to stream-stage and recharge variations. I. Analytical step-response

functions. J Hydrol,230(2000) 192210.

[18] PY Polubarinova-Kochina Theory of groundwater movement, Princeton University Press, N.J. 1962.

[19] Rai, S.N. Singh, R.N., On the prediction of groundwater mound due to transient recharge from rectangular area.

Water ResourManag,10(1986) 189198.

[20]Upadhyaya, A., Chauhan H.S., Water table rise in sloping aquifer due to canal seepage and constant recharge. J IrrgDrain Eng,128(2002) 160167.

[21]Verhoest, N.E.C.,Troach,P.A., Some analytical solution of the linearized Boussinesq equation with recharge for a

sloping aquifer. Water Resour Res,36(3) (2000)793800.

-0.8

0

0.8

1.6

0 50 100 150FlowRateqx

=L

(m/h)

Time t (h)

=0.108h1

=0.072h1

=0.054h1

=0.036h1

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Areal test of the effectiveness of an approach to uncertainty is the capability to solveproblems which involve different facets of uncertainty. Fuzzy logic has a much higher

problem solving capability than standard probability theory. Most importantly, it opens thedoor to construction of mathematical solutions of computational problems which are stated

in a natural language. The applications which may be generated from or adapted to fuzzy

logic are wide-ranging and provide the opportunity for modelling under conditions which

are inherently imprecisely defined, despite the concerns of classical logicians (e.g. see [8],Chapter 6 of [10], [13], [17-19], [20] and its relevant references, [21-24], etc).

The methods of assessing the individuals performance usually applied in practice arebased on principles of the bivalent logic (yes-no). However these methods are not probably

the most suitable ones. In fact, fuzzy logic, due to its nature of including multiple values,

offers a wider and richer field of resources for this purpose. In this paper we shall use fuzzy

logic in developing a new general method for assessing the skills of groups of individualsparticipating in any human activity. The rest of the paper is organized as follows: In the

next section we develop our new assessment method. In section three we present two real

applications illustrating the importance of our method in practice. Finally the last section is

devoted to conclusions and discussion for the future perspectives of research on this area.For general facts on fuzzy sets we refer freely to the book [10].

2. The Fuzzy Model

Let us consider a group, say H, of nindividuals, where nis a positive integer, participatingin a human activity (e.g. problem-solving, decision making, football match, a chess

tournament, etc). Further, let U= {A, B, C, D, F} be a set of linguistic labels characterizingthe individuals performance with respect to the above activity, where A characterizes an

excellent performance, B a very good, C a good, D a mediocre and F an unsatisfactory

performance respectively. Obviously, the above characterizations are fuzzy depending on

the users personal criteria, which however must be compatible to the common logic, inorder to be able to model the real situation in a worthy of credit way.

The classical way for assessing the total groups performance with respect to the

corresponding activity is to express the levels of the individuals performance in numerical

values and then to calculate the mean of their performance in terms of these values (mean

groups performance).

Here, we shall use principles of fuzzy logic in developing an alternative method ofassessment, according to which the higher is an individuals performance, the more itscontribution to the groups total performance (weighted groups performance). For this,

we are going to representHas a fuzzy subset of U.In fact, if nA, nB.nC, nDand nFdenote the

number of the individuals of H that had demonstrated an excellent, very good, good,mediocre and unsatisfactory performance respectively at the game, we define the

membershipfunction (We recall that the methods of choosing the membership function are usuallyempiric, based either on the common logic (as it happens in our case) or on the data of experiments made

on a representative sample of the population that we study. The proper choice of the membership functionis a necessary condition for developing a worthy of credit model of the corresponding situation using

principles of fuzzy logic.)

m:U [0, 1] as follows:

1, 80%

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x

x

x

x

x

if n n n

if n n n

y m x if n n n

if n n n

if n n

(1)

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for

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Therefore formulas (2) are transformed into the following form:

Normalizing our fuzzy data by dividing each m(x),xU, with the sum of all membershipdegrees we can assume without loss of the generality thaty1+y2+y3+y4+y5= 1(We recall that, although probabilities and fuzzy membership degrees are both taking values in theinterval [0, 1], they are distinctand different to each other notions. For example, the expression The

probability for Mary to be tall is 85%, means that, although Mary is either tall or low in stature, she is

very possibly tall. On the contrary the expression The membership degree of Mary to be tall is 0.85,

simply means that Mary can be characterized as rather tall. A.characteristic difference (but not the onlyone) is that, while the sum of probabilities of all the singleton subsets (events) of Uequals 1, this is not

necessary to happen for the membership degrees.)Therefore we can write:

withyi=

Ux

i

xm

xm

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)(.

But, 0 (y1-y2)2=y1

2+y2

2-2y1y2,thereforey1

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2+y3

2 2y1y3, with the equality holding if,and only if, y1=y3 and so on. Hence it is easy to check that (y1+y2+y3+y4+y5)

2 5(y1

2+y2

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Buty1+y2+y3+y4+y5 =1, therefore1 5(y1

2+y2

2+y3

2+y4

2+y5

2) (4),with the equality holding if, and only if,

y1=y2=y3=y4=y5=5

1

Then the first of formulas (3) gives that xc =2

5 . Further, combining the inequality (4)

with the second of formulas (3), one finds that 1 10yc, oryc 10

1 . Therefore the unique

minimum foryccorresponds to the centre of gravity Fm(2

5,10

1 ).

The ideal case is wheny1=y2=y3=y4=0andy5=1.Then from formulas (3) we get thatxc=

2

9 andyc=2

1 .Therefore the centre of gravity in this case is the point Fi(2

9,2

1 ).

On the other hand, in the worst casey1=1andy2=y3=y4= y5=0. Then by formulas (3), wefind that the centre of gravity is the point Fw(

2

1,2

1 ).

Therefore, the area where the centre of gravity Fclies is represented by the triangleFwFmFi of Figure 2. Then from elementary geometric considerations it follows that the

greater is the value ofxcthe better is the corresponding groups performance.

(3)

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2

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c

x y y y y y

y y y y y y

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Also, for two groups with the same xc 2,5, the group having the centre of gravitywhich is situated closer to Fiis the group with the higheryc; and for two groups with the

same xc

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On the contrary, the usual method of scoring in a pairs competition is in match points.Each pair is awarded two match points for each pair who scored worse than them on eachgames session (hand), and one match point for each pair who scored equally. The total

number of match points scored by each pair over all the hands played is calculated and it isconverted to a percentage.

However, IMPs can also used as a method of scoring in pair events. In this case the

difference of each pairs IMPs is usually calculated with respect to the mean number of

IMPs of all pairs.For the fundamentals and the rules of bridge, as well as for the conventions usually

played between the partners we refer to the famous book [9] of Edgar Kaplan

(1925-1997), who was an American bridge player and one of the principal contributors to

the game. Kaplans book was translated in many languages and was reprinted many times

since its first edition in 1964. There is also a fair amount of bridge-related information onthe Internet, e.g. seeweb sites [2], [4], etc.

The Hellenic Bridge Federation (HBF) organizes, on a regular basis, simultaneous

bridge tournaments (pair events) with pre-dealt boards, played by the local clubs in several

cities of Greece. Each of these tournaments consists of six in total events, played in a

particular day of the week (e.g. Wednesday), for six successive weeks. In each of theseevents there is a local scoring table (match points) for each participating club, as well as a

central scoring table, based on the local results of all participating clubs, which are

compared to each other. At the end of the tournament it is also formed a total scoring tablein each club, for each player individually. In this table each players score equals to the

mean of the scores obtained by him/her in the five of the six in total events of thetournament. If a player has participated in all the events, then his/her worst score is

dropped out. On the contrary, if he/she has participated in less than five events, his/her

name is not included in this table and no possible extra bonuses are awarded to him/her.

In case of a pairs competition with match points as the scoring method and according to

the usual standards of contract bridge, one can characterize the players performance,according to the percentage of success, say p, achieved by them, as follows:Excellent (A), if p > 65%.

Very good (B), if 55% < p 65%.Good (C), if 48% < p 55%.Mediocre (D), if 40% p 48%.Unsatisfactory (F), if p < 40 %. (In an analogous way one could characterize the players

performance in bridge games played with IMPs, with respect to the VPs gained.)Our application presented here is related to the total scoring table of the players of a

bridge club of the city of Patras, who participated in at least five of the six in total events of

a simultaneous tournament organized by the HBF, which ended on February 19, 2014 (see

results in [7]). Nine men and five women players are included in this table, who obtainedthe following scores. Men: 57.22%, 54.77%, 54.77%, 54.35%, 54.08%, 50.82 %, 50.82%,

49.61%, 47.82%. Women: 59.48%, 54.08%, 53.45%, 53.45%, 47.39%. These results givea mean percentage of approximately 52.696% for the men and 53.57% for the women

players. Therefore the women demonstrated a slightly better mean performance than themen players, their difference being only 0.874%.

The above results are summarized in Table 1, the last column of which contains the

corresponding membership degrees calculated with respect to the membership functiondefined by the formula (1) of the previous section. For example,

in the case of men players we have n = 9, 1% n = 0.09, 20% n = 1.8, 50% n = 4.5, 80% n=7.2, nA= nF= 0, nB= nD= 1, nC= 7, which give that m(A) = m(B) )= m(F) = 0, m(B) =

m(D) = 0.25 and m(C)= 0.75

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Men

% Scale Performance Number

of players

m(x)

>65% A 0 0

55-65% B 1 0.25

48-55% C 7 0.75

40-48% D 1 0.25

65% A 0 0

55-65% B 1 0.25

48-55% C 3 0.75

40-48% D 1 0.25

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Therefore, the kind of problem will dictate the type of cognitive skill necessary to solve

the problem: linguistic skills are used to read about a certain problem and debate about it,memory skills to recall prior knowledge and so on. Depending on the knowledge and

thinking skills possessed by a problem solver, what could be a problem for one might not

be a problem for some body else. Perhaps Martinezs [11] definition carries the modern

message about PS: PS can be defined simply as the pursuit of a goal when the path tothat goal is uncertain. In other words, its what you do when you dont know what youredoing.

Mathematics by its nature is a subject whereby PS forms its essence. In an earlier paper[22] we have examined the role of the problem in learning mathematics and we have

attempted a review of the evolution of research on PS in mathematics education from its

emergency as a self sufficient science at the 1960s until today.

While early work on PS focused mainly on analysing the PS process and on describing theproper heuristic strategies to be used in each of its stages, more recent investigations have

focused mainly on solvers behaviour and required attributes during the PS process; e.g.see Multidimensional PS Framework (MPSF) of Carlson & Bloom [3]. More

comprehensive models for the PS process in general (not only for mathematics) were

developed by Sternberg & Ben-Zeev [16], by Schoenfeld [15] (PS as a goal-orientedbehaviour), etc. In earlier papers we have used basic elements of the finite Markov chains

theory, as well as principles of the fuzzy logic (e.g. see [20] and its relevant references) in

an effort to develop mathematical models for a better description and understanding of the

PS process. Here, applying the results obtained in the previous section, we shall provide a

new assessment method of the students PS skills.In our will to explore the effect of the use of computers as a tool in solving mathematical

problems we performed during the academic year 2012-13 a classroom experiment, in

which the subjects were students of the School of Technological Applications (prospectiveengineers) of the Graduate Technological Educational Institute (T. E. I.) of Western

Greece attending the course Higher Mathematics I(The course involves Complex Numbers,Differential and Integral Calculus in one variable, Elementary Differential Equations and Linear

Algebra.)of their first term of studies. The students, who had no previous computerexperience apart from the basics learned in High School, where divided in two groups. In

the control group (100 students) the lectures were performed in the classical way on the

board, followed by a number of exercises and examples connecting mathematics with realworld applications and problems. The students participated in solving these problems. Thedifference for the experimental group (90 students) was that part (about the 1/3) of the

lectures and the exercises were performed in a computer laboratory. There the instructorused the suitable technological tools (computers, video projections, etc) to present the

corresponding mathematical topics in a more live and attractive to students way, while

the students themselves, divided in small groups, used standard mathematical software tosolve the problems with the help of computers.

At the end of the term all students participated in the final written examination for the

assessment of their progress. The examination involved a number of general theoretical

questions and exercises covering all the topics taught and three simplified real world

problems (see Appendix) requiring mathematical modelling techniques for their solution(the time allowed was three hours). We marked the students papers separately for the

questions and exercises and separately for the problems. The results of this examination

are summarized in Tables 2 {theoretical questions and exercises) and 3 (real worldproblems) , where the corresponding membership degrees were calculated in terms of the

membership function (1), as it has been already shown in application 3.1.

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Experimental group

% Scale Performance Number of students M(x)

89-100 A 0 0

77-88 B 17 0.25

65-76 C 18 0.25

53-64 D 25 0.5

Less than 53 F 30 0.5

Total 90 1.5

Control group


89-10 A 0 0

77-88 B 18 0.25

65-76 C 20 0.25

53-64 D 30 0.5


Total 100 1.5

Table 2: Theoretical questions and exercises

Experimental group

% Scale Performance Number of

Students

M(x)

89-100 A 3 0.25

77-88 B 21 0.5

65-76 C 28 0.5

53-64 D 22 0.5


Total 90 2

Control group


89-100 A 1 0

77-88 B 10 0.25

65-76 C 37 0.5

53-64 D 31 0.5


Total 100 1.75Table 3: Real world problems

Normalizing the membership degrees and applying formulas (3), it is easy to check from

Table 2 that the two groups demonstrated identical performance with respect to theirstudents replies to the theoretical questions and their solutions of the exercises. In fact, forboth groups we find that

1 5( 0 . 5 3 * 0 . 5 5 * 0 . 2 5 7 * 0 . 2 5 ) 1 . 6 6 7

2 * 1 . 5 3c

x

andyc=

2 2 2 2

2

1 0 .5 6 9[ ( 0 .5 ) ( 0 .5 ) ( 0 .2 5 ) ( 0 .2 5 ) ] 0 .1 2 6

4 .52 * (1, 5 )

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(third case of our criterion).

Similarly from Table 3 we find1 10

(0.25 3* 0.5 5* 0.5 7 * 0.5 9 * 0.25) 2.52 * 2 4

cx for the experimental and

1 6.25(0.5 3*0.5 5*0.5 7 *0.25) 1.786

2*1.75 3.5c

x

for the control group, which means that the experimental group demonstrated a better

performance with respect to the solution of the real problems than the control group.

Further, it is worthy to notice that the experimental groups performance was

significantly better with respect to the solution of the real world problems than with

respect to the theoretical questions and exercises (1.667

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5. References

[1] Bridge Base Online (BBO), www.bridgebase.com/index.php?d=y , accessed on

March 29, 2014

[2]

Bridge rules and variations, www.pagat.com/boston/bridge.html , accessed on March29, 2014

[3] Carlson, M. P. & Bloom, I., The cyclic nature of problem solving: An emergentmultidimensional problem-solving framework, Educational Studies in Mathematics,

58 (, (2005), 45-75.

[4] Contract bridge, http://en.wikipedia.org/wiki/contract_bridge, accessed on March 29,2014

[5] DOrsi, E. (chair) et al., The New WBF IMP to VP Scales, Technical Report of

WBF Scoring,www.worldbridge.org/Data/Sites/1/media/documents/regulations/WBFAlgorithms.p

df, accessed on March 29, 2014

[6] Green, A. J. K. & Gillhooly, K., Problem solving. In:Braisby, N. & Gelatly, A.(Eds.), Cognitive Psychology (2005), Oxford University Press, Oxford.

[7] Hellas Bridge Federation(in Greek), www.hellasbridge.org, accessed on March 31,

2014[8] Jelodar, Sh. S., Allahviranloo, T. Abbasbandy, Application of Fuzzy Expansion

Methods for Solving Fuzzy Fredhom-Voltera Integral Equations of the First Kind,International Journal of Mathematical Modelling & Computations, 1(2013), 59-70

[9] Kaplan, E., Winning Contract Bridge (2010), Courier Dover Publications, N. Y.

[10]Klir, G. J. & Folger, T. A., Fuzzy Sets (1988), Uncertainty and Information (1988),Prentice-Hall, London.

[11]Martinez, M., What is meta cognition? Teachers intuitively recognize the

importance of meta cognition, but may not be aware of its many dimensions, Phi

Delta Kappan,87(9) (2007), 696-714.[12]Polya, G., How to solve it (1945), Princeton University Press, Princeton.

[13]Sajadi, S., Amirfakhrian, M., Fuzzy K-Nearest Neighbor Method to Classify Data in

A Closed Area, International Journal of Mathematical Modelling & Computations,2(2013), 109-114

[14] Schoenfeld, A., The wild, wild, wild, wild world of problem solving: A review of

sorts, For the Learning of Mathematics, 3 (1983), 40-47.

[15]Schoenfeld, A., How we think: A theory of goal-oriented decision makingand its educational applications (2010),New York: Routledge.

[16]Sternberg, R. J. & Ben-Zeev, T., Complex cognition: The psychology of human

thought (2001), Oxford University Press, Oxford.[17]Subbotin, I. Ya. Badkoobehi, H., Bilotckii, N. N., Application of fuzzy logic to

learning assessment. Didactics of Mathematics: Problems and Investigations, 22

(2004), 38-41.

[18]Van Broekhoven, E. & De Baets, B., Fast and accurate center of gravitydefuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions, Fuzzy

Sets and Systems, 157 (7) (2006), 904-918 .

[19]Voskoglou, M. Gr., Fuzzy Sets in Case-Based Reasoning, Fuzzy Systems andKnowledge Discovery, IEEE Computer Society, Vol. 6 (2009), 252-256.

[20]Voskoglou, M. Gr., Stochastic and fuzzy models in Mathematics Education, Artificial

Intelligence and Management (2011), Lambert Academic Publishing, Saarbrucken,

Germany.

[21]Voskoglou, M. Gr., Fuzzy Logic and Uncertainty in Mathematics Education,International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 1

(2011), 45-64.[22]Voskoglou, M. G., Problem Solving from Polya to Nowadays: A Review and Future

Perspectives. In: R. V. Nata (Ed.), Progress in Education, Vol. 22 (2011), Chapter 4,

65-82, Nova Publishers, N.Y.

[23]Voskoglou, M. Gr., Measuring students modelling capacities: A fuzzy approach,

Iranian Journal of Fuzzy Systems, 8(3) (2011), 23-33.

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[24]Voskoglou, M. Gr. & Subbotin, I. Ya., Fuzzy Models for Analogical Reasoning,International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 2

(2012), 19-38

[25]Zadeh, L. A., Fuzzy Sets, Information and Control, 8 (1965), 338- 353.

Appendix: The problems given for solution to students in our classroom experiments

Problem 1: We want to construct a channel to run water by folding the two edges of an

orthogonal metallic leaf having sides of length 20cm and 32 cm, in such a way that they

will be perpendicular to the other parts of the leaf. Assuming that the flow of the water isconstant, how we can run the maximum possible quantity of the water?

Remark: The correct solution is obtained by folding the edges of the longer side of the leaf.

Some students solved the problem by folding the edges of the other side and failed torealize that their solution was wrong (validation of the model).

Problem 2:Let us correspond to each letter the number showing its order into the alphabet(A=1, B=2, C=3 etc). Let us correspond also to each word consisting of 4 letters a 2X2

matrix in the obvious way; e.g. the matrix

513

1519corresponds to the word SOME.

Using the matrix E=

711

58 as an encoding matrix how you could send the message

LATE in the form of a camouflaged matrix to a receiver knowing the above process andhow he (she) could decode your message?

Problem 3: The population of a country is increased proportionally. If the population is

doubled in 50 years, in how many years it will be tripled?

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ved: 18 July 2013

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Formulation

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rmala P. Ratc

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iversity, Anna

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the operati

cheduling prt-term hydrot

ixed Head

amalai Naga

milnadu, Indi

lve the fixed

blem, watert is modeled

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and thermalance of the

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n planning o

blem is one oermal sched

,

a.

f an

f theling

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S. Kavitha&Nirmala P. Ratchagar/ , 04 - 03 (2014) 213 - 222

214

involves the hour-by-hour scheduling of all generating units on a system to achieveminimum generation cost. The objective of fixed head hydrothermal scheduling is tominimize the fuel cost of thermal generating units by determining the optimal hydro andthermal generation schedule subject to satisfying water availability constraint and loaddemand in each interval of the scheduling horizon. The head of the reservoir is assumed tobe constant for the hydro plants having reservoirs with large capacity. The hydraulicconstraint imposed in the problem formulation is that the total volume of water discharged

in the scheduling horizon must be exactly equal to the defined volume. The operationalcost of hydroelectric units is insignificant because the source of hydro power is the naturalwater resources.Numerous mathematical methods for the optimal resource allocation problems have beendeveloped in the literature [6, 7]. Different mathematical methods for hydrothermalscheduling problem have been reported in the literature. In [12] , an iterative techniquebased on computing LU factors of the Jacobian with partial pivoting for solvinghydrothermal scheduling problem has been developed. A Lagrangian multiplier methodfor optimal scheduling of fixed head hydro and thermal plants is reported in [5]. Thismethod linearizes the coordination equation and solves for the water availability constraintseparately from unit generations. Dynamic programming approach [10] has beenimplemented for the solution of hydrothermal scheduling problem. The disadvantages ofdynamic programming method are computational and dimensional requirements growdrastically with increase in system size and planning horizon. Lagrangian relaxationtechniques have been proposed in the literature to solve hydrothermal coordinationproblem [11, 4]. A Lagrangian relaxation technique solves the dual problem of the originalhydrothermal coordination problem. However the perturbation procedures are required toobtain primal feasible solution. These perturbation procedures may deteriorate theoptimality of the solution obtained.Recently, as an alternative to the conventional mathematical approaches, the heuristicoptimization techniques such as simulated annealing [1], genetic algorithm [9], artificialimmune algorithm [2] have been used to solve fixed head hydrothermal schedulingproblem. Heuristic methods use stochastic techniques and include randomness in movingfrom one solution to the next solution. Due to the random search nature of the algorithm,these methods provide feasible optimal solution for the optimization problems. Neuralnetwork based approach has been developed for scheduling thermal plants in coordinationwith fixed head hydro units [3]. This method provides near-optimal solution for hydro

thermal scheduling problem. An integrated technique of predator-prey optimization andPowells method for optimal operation of hydrothermal system has been reported in [8].In this hybrid method, predator-prey optimization is used as a base level search in theglobal search space and Powells method as a local search technique.In this paper, a new deterministic method based on Lagrangian multiplier and Newtonmethod is developed for the solution of short-term fixed head hydrothermal schedulingproblem. The scheduling period is divided into a number of subintervals each having aconstant load demand. The proposed approach has been validated by applying it to threetest systems.

2. Problem Formulation

The basic problem is to find the real power generation of committed hydro and thermalgenerating units in the system as a function of time over a finite time period from 1 to T.

The goal is to minimize the total fuel cost required for the thermal generation in thescheduling period.

)(PFtMinimize Tik

T

1k

NT

1i

ikk

(1)

whereFik(PTik) is cost function of each thermal generating unit in interval k, which is

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expressed by iTiki2TikiTikik cPbPa)(PF

T- number of periods for dividing the scheduling time horizonNT- number of thermal generators

tk- time in interval kPTik- power generation level of ith thermal generating unit in interval kai, biand ciare the fuel cost coefficients of ith thermal generating unitSubject to

(i)

Power balance constraint

0PPPC Dk

NH

1i

Hik

NT

1i

Tikk

for k 1, 2, , T (2)

wherePHik- power generation level of ith hydro generating unit in interval kPDk- Total generation demand in interval k.NH - number of hydro generators

(ii) Water availability constraint

iiHikik

T

1k

ki S-V)(PqtW

for i= 1, 2, , NH (3)

whereqik is the discharge rate of hydro unit i during the kth interval, which is expressedby iHiki

2HikiHikik PP)(Pq .

Vi is the pre-specified volume of water available for hydro unit i during the schedulingperiod. Siis the total spillage discharge of ith hydro unit during the scheduling period and

the spillage discharge is not used for power generation. iii and,, are the discharge

coefficients of ith hydro unit withmaxTiTik

minTi PPP (4)

maxHiHik

minHi PPP (5)

where maxTiminTi P,P are the minimum and maximum generation limits of ith thermal unit

and maxHiminHi P,P are the minimum and maximum generation limits of ith hydro unit.

3. ProposedMethodology

The lagrangian equation for hydrothermal scheduling is

))S-V((WC),L(P, iii

NH

1i

i

T

1k

kk

(6)

wherekis the Lagrangian multiplier for power balance constraint in kth interval and iisthe lagrangian multiplier for hydro unit i. Substituting eqns. (1),(2) and (3) in eqn (6)gives

))S-V()(Pqt(

)PPP()(PFt),L(P,

iiHikik

T

1k

k

NH

1i

i

T

1k

Dk

NH

1i

Hik

NT

1i

TikkTik

T

1k

NT

1i

ikk

(7)

The minimum value is obtained by partially differentiating the Equation (7) with respect toPTik, PHik, k,iand equating to zero

i.e 0P

L

Tik

(8)

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i.e 0P

C

P

)(PFt

Tik

kk

Tik

Tikikk

(9)

Equation (9) gives

kiTikik )bP(2at (10)

i.e 0P

L

Hik

(11)

therefore0))S-V((W

P iii

Hikik

(12)

Equation (12) gives

kiHikiik ]P[2t (13)

i.e 0L

k (14)

therefore 0Ck (15)

Equation (15) gives

Dk

NH

1i

Hik

NT

1i

Tik PPP

(16)

i.e 0

L

i

(17)

therefore 0)S-V(W iii (18)

Equation (18) gives

iiiHiki2Hik

T

1k

ik S-V]PP[t

(19)

From the Equation (10), we get

i

i

ik

i

ikTik

2a

b

2a

1

2a

bP

iikTik BAP for i = 1, 2, , NT, k= 1, 2, , T (20)

where i

i

iii 2a

b

B;2a

1

A

From the Equation (13), we get

i

i

ii

k

ii

iikHik

2

2

2

P

iii

kHik DC

P for i = 1, 2, , NH, k= 1, 2, , T (21)

wherei

ii

ii

2

D;

2

1C

Substituting Equations (20) and (21) in Equation (16) gives

Dk

NH

1i

ii

ikii

NT

1i

k PD

C)BA(

for k= 1, 2, , T

therefore

NT

1i

NH

1i i

ii

NT

1i

NH

1i

iiDk

k

CA

DBP

for k= 1, 2, , T (22)

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SubstitutingEquation (22) in Equation (21) gives

iNT

1i

NH

1i i

i

i

NT

1i

NH

1i

iiDk

i

iHik D

CA

DBP

CP

for i =1, 2, , NH, for k= 1, 2, , T

(23)

Substituting (23) in (19), we get

iiiiNT

1i

NH

1ii

ii

NT

1i

NH

1i

iiDk

i

ii

2

iNT

1i

NH

1i i

ii

NT

1i

NH

1i

iiDk

i

ii

T

1k

k

S-V)D

CA

DBP

C

D

CA

DBP

C(t

for i =1, 2, , NH

(24)

By expanding the Equation (24)

iiiiiNT

1i

NH

1i i

ii

NT

1i

NH

1i

iiDk

i

ii

NT

1i

NH

1i i

ii

NT

1i

NH

1i

iiDk

i

ii2i

2

NT

1i

NH

1i i

ii

NT

1i

NH

1i

iiDk

2i

2i

i

T

1k

k

S-V)D

CA

DBP

C

CA

DBP

D2CD

CA

DBP

C(t

for i =1, 2, , NH

(25)

Simplifying the Equation (25)

0)S-(V)DD24(

CA

DBP

CA

D2C

C

DBP

C

iiiii2ii

2NT

1i

NH

1i i

ii

T

1k

NT

1i

NH

1i

iiDk

NT

1i

NH

1i i

ii

i

iii

i

ii

2T

1k

NT

1i

NH

1i

iiDk2i

2ii

for i =1, 2, , NH

(26)

The Equation (26) is solved by Newtons method.The solution of these equations gives thevalues of 1, 2 , , n . The detailed procedure for solving the simultaneous equationswith two variables 1 and 2by Newtons method is given below:Let f (1, 2) = 0, and g(1, 2) = 0 (27)

Take 0201and be the initial approximate solution for the above equation. The actual

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solution is given by m)(andh)( 0201 ,where h is an incremental value of 1 and m is

an incremental value of 2.

Therefore, 0m)h,(f 0201 (28)

0m)h,g( 0201 (29)

Expanding equations (28) and (29) by Taylor series,

0)],([fm)],([fh),(f0201

20201

10201

(30)

0)],([g

m)],([g

h),(g 0201

2

02

01

1

02

01

(31)

Equations (30) and (31) can be written as

0)m(f)h(ff 02010 (32)

0)m(g)h(gg 02010 (33)

Equations (32) and (33) are solved for h and m using determinants,

i.eD

Dmand

D

Dh

21

(34)

where

0201

0201

)(g)(g

)(f)(fD ;

020

0201

)(gg

)(ffD ;

001

0012

g)(g

f)(fD

By using the incremental values, the new values of 1 and 2are obtained by

;h o1new1 m

o2

new2 (35)

Nownew1 and

new2 are taken as initial values and above process is repeated till the

convergence criterion is satisfied.

The computational procedure for implementing the proposed method for the solution offixed head hydrothermal scheduling problemis given in the following steps:

Step 1: Calculate the initial generation of thermal, hydro plants and initial kfrom thefollowing equations

T,...2,1,kNT,,...2,1,ifor

NHNT

PP DkTik

T,...2,1,kNH,,...2,1,iforNHNT

PP DkHik

T,...2,1,kNT,,...2,1,iforbP2a iTikik

Step 2: Determine the initial values of 0i using Equation (13)

i.e NH,...2,1,iforP2

iHiki

k0i

Step 3: Substitute the values of 0i in Equation (26).

Step 4: Calculate the incremental values of iusingEquation (34).Step 5: Calculate thenewivalues using Equation(35).Step 6: Substitute the new ivalues in Equation(26) and then go to Step 4. This iterativeprocedureis continued till the difference between two consecutive iterationsis less than specified tolerance.Step 7: Calculate kvaluesfrom the Equation (22)bysubstitutingtheoptimal values of i.Step 8: Determine the optimal generation schedule of thermal and hydro plants usingeqns. (20) & (21).

4. Numerical Examples and Results

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Amathematical approach developed in this paper has been implemented on three fixedhead hydrothermal scheduling problems. The program is developed using MATLAB 7.The system data and results obtained through the proposed approach are given below

Table 1 Optimal generation schedule for Example 1

Interval k Demand

PDk

Water discharge

q1k

Hydro generation

PH1k

Thermal generation

PT1k1 455 101.9329 234.2742 220.7258

2 425 101.0896 231.8794 193.1206

3 415 100.8105 231.0812 183.9188

4 407 100.5879 230.4426 176.5574

5 400 100.3937 229.8838 170.1162

6 420 100.9500 231.4803 188.5197

7 487 102.8422 236.8285 250.1715

8 604 106.2529 246.1679 357.8321

9 665 108.0848 251.0372 413.9628

10 675 108.3886 251.8354 423.1646

11 695 108.9992 253.4319 441.5681

12 705 109.3059 254.2301 450.7699

13 580 105.5423 244.2522 335.747814 605 106.2827 246.2477 358.7523

15 616 106.6104 247.1258 368.8742

16 653 107.7215 250.0793 402.9207

17 721 109.7988 255.5073 465.4927

18 740 110.3874 257.0240 482.9760

19 700 109.1524 253.8310 446.1690

20 678 108.4799 252.0749 425.9251

21 630 107.0292 248.2433 381.7567

22 585 105.6899 244.6513 340.3487

23 540 104.3705 241.0592 298.9408

24 503 103.3007 238.1057 264.8943

Problem- 1

Minimize

24

1k

T1k2T1kT11 373.7P9.606P0.001991)(PF dollars

Subject to satisfying power demand in each interval and total water discharge during theentire scheduling horizon should be equal to the given water availability.Total volume of water available for discharge (24 hours) V = 2559.6 M cubic ft

Water discharge 61.53P0.009079-P0.0007749)(Pq H1k2H1kH1k1k M cubic ft. per

hour ;Total spillage S= 25.596 M cubic ft/24 hoursThe optimal generation schedule obtained through the proposed method is given inTable 1. The solution converged in fifth iteration. The optimal value of = 29.61852312and total amount of water utilized for hydro power generation exactly satisfied wateravailability constraint. The total fuel cost of thermal power generation is $ 92097.74.

Problem- 2

Minimize

24

1k

T1k2T1kT11 15P0.3P0.01)(PF dollars

Subject to satisfying power demand in each interval and total water discharge during the

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entire scheduling horizon should be equal to the given water availability constraints.Total volume of water available for Hydro plant 1 (24 hours) V1= 25 M cubic ftTotal volume of water available for Hydro plant 1 (24 hours) V2= 35 M cubic ft


Intervalk

DemandPDk

Water discharge Hydro generation Thermal generationPT1k

q1k q2k PH1k PH2k

1 30 0.7714 0.9543 18.4792 9.1003 2.4205

2 33 0.8105 1.0279 19.7041 10.2892 3.0067

3 35 0.8367 1.0772 20.5207 11.0817 3.3976

4 38 0.8760 1.1513 21.7456 12.2706 3.9838

5 40 0.9023 1.2009 22.5622 13.0631 4.3746

6 45 0.9684 1.3253 24.6038 15.0446 5.3517

7 50 1.0349 1.4505 26.6453 17.0260 6.3287

8 59 1.1556 1.6780 30.3201 20.5925 8.0874

9 61 1.1826 1.7288 31.1367 21.3851 8.4782

10 58 1.1421 1.6526 29.9118 20.1962 7.8920

11 56 1.1152 1.6019 29.0951 19.4037 7.5012

12 57 1.1286 1.6272 29.5034 19.7999 7.6966

13 60 1.1691 1.7034 30.7284 20.9888 8.2828

14 61 1.1826 1.7288 31.1367 21.3851 8.4782

15 65 1.2368 1.8310 32.7699 22.9702 9.2599

16 68 1.2776 1.9079 33.9948 24.1591 9.8461

17 71 1.3186 1.9851 35.2197 25.3479 10.4324

18 62 1.1961 1.7543 31.5450 21.7814 8.6737

19 55 1.1018 1.5766 28.6868 19.0074 7.3058

20 50 1.0349 1.4505 26.6453 17.0260 6.3287

21 43 0.9419 1.2754 23.7872 14.2520 4.9609

22 33 0.8105 1.0279 19.7041 10.2892 3.0067

23 31 0.7845 0.9788 18.8875 9.4966 2.6159

24 30 0.7714 0.9543 18.4792 9.1003 2.4205

Water discharge equations

2.0P0.03P0.00005)(Pq H1k2H1kH1k1k M cubic ft. per hour


Total spillageS1= 0.25 M cubic ft/24 hours, S2= 0.35 M cubic ft/24 hours

The optimal generation schedule of hydro and thermal plants obtained through theproposed method is given in Table 2. The solution converged at fifth iteration. Theoptimal values of 1 = 95.717727 and 2 = 49.311017. The total fuel cost of thermalpower generation is $ 821.32.

Problem- 3Minimize F1(PT1)+ F2(PT2) where

24

1k


24

1k


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Subject to satisfying power demand in each interval and total water discharge during theentire scheduling horizon should be equal to the given water availability constraints.Total volume of water available for Hydro plant 1 (24 hours) V1= 2500 M cubic ftTotal volume of water available for Hydro plant 1 (24 hours) V2= 2100 M cubic ft


Interval

k

Demand

PDk

Water discharge Hydro

generation

Thermal generation

q1k q2k PH1k PH2k PT1k PT2k

1 400 64.8580 21.3191 182.0811 32.6776 75.2100 110.0313

2 300 57.6834 9.5683 163.2297 13.9900 60.0679 62.7123

3 250 54.1537 3.7873 153.8040 4.6462 52.4969 39.0529

4 250 54.1537 3.7873 153.8040 4.6462 52.4969 39.0529

5 250 54.1537 3.7873 153.8040 4.6462 52.4969 39.0529

6 300 57.6834 9.5683 163.2297 13.9900 60.0679 62.7123

7 450 68.5029 27.2888 191.5068 42.0214 82.7810 133.6907

8 900 103.0338 83.8446 276.3381 126.1156 150.9203 346.6260

9 1230 130.3323 128.5549 338.5477 187.7847 200.8891 502.7785

10 1250 132.0405 131.3527 342.3180 191.5222 203.9175 512.2423

11 1350 140.6736 145.4921 361.1694 210.2098 219.0596 559.561212 1400 145.0477 152.6562 370.5951 219.5536 226.6306 583.2207

13 1200 127.7816 124.3772 332.8923 182.1784 196.3465 488.5828

14 1250 132.0405 131.3527 342.3180 191.5222 203.9175 512.2423

15 1250 132.0405 131.3527 342.3180 191.5222 203.9175 512.2423

16 1270 133.7549 134.1604 346.0883 195.2597 206.9459 521.7061

17 1350 140.6736 145.4921 361.1694 210.2098 219.0596 559.5612

18 1470 151.2359 162.7914 383.7911 232.6349 237.2301 616.3439

19 1330 138.9347 142.6441 357.3991 206.4723 216.0312 550.0974

20 1250 132.0405 131.3527 342.3180 191.5222 203.9175 512.2423

21 1170 125.2446 120.2221 327.2369 176.5721 191.8039 474.3871

22 1050 115.2350 103.8280 304.6152 154.1470 173.6334 417.6044

23 900 103.0338 83.8446 276.3381 126.1156 150.9203 346.6260

24 600 79.6677 45.5750 219.7839 70.0528 105.4941 204.6692

Water discharge equations



Total spillageS1= 25 M cubic ft/24 hours, S2= 21 M cubic ft/24 hours

The optimal generation schedule of example 3 is given in Table 3. The solutionconverged at fifth iteration. The optimal values of 1= 9.2967 and 2= 5.6269. The totalfuel cost of thermal power generation is $ 47985.76.

5. Conclusion

A novel mathematical approach for the solution of fixed head hydrothermal schedulingproblem is presented in this paper. The proposed method is implemented with test systems.The practical constraints of fixed head hydrothermal scheduling such as water availabilityconstraints, waterspillage,power balance constraint in each interval and generation limits

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of hydro and thermal units are taken into account in the problem formulation. Numericalresults show that the proposed method has the ability to determine the global optimalsolution and the method requires lesser number of iterations for the convergence.

References

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[2]

BasuM., Artificial immune system for fixed head hydrothermal power system, Energy, 36(2011) 606-612.[3] BasuM., Hopfield neural networks for optimal scheduling of fixed head hydrothermal power systems,

Electric power system research, 64(2003) 11-15.[4]

GuanX.,LuhP.B.,ZhangL., Nonlinear approximation in Lagrange relaxation based algorithm forhydrothermal scheduling, IEEE transactions on power systems, 10(1995) 772-778.

[5] HalimA., RashidA.,NorK.M., An efficient method for optimal scheduling of fixed head and thermal plants,IEEE transactions on power systems, 6(1991) 632-636.

[6]

Masoumzadeh A., Moradmand Rad N., Optimal resource allocation in DEA with integer variables,International Journal of Mathematical modelling& computations, 1(2011) 251-256.

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Mamizadeh-Chatghayeh S., Cost and allocative efficiency in buyer-supplier, International Journal ofMathematical modelling& computations, 2(2012) 151-158.

[8] NarangN., DhillonJ.S., KothariD.P., Multiobjective fixed head hydrothermal scheduling using integratedpredator-prey optimization and powell search method, Energy, 47(2012) 237-252.

[9]

SasikalaJ.,RamaswamyM., Optimal gamma based fixed head hydrothermal scheduling using geneticalgorithm, Expert system with applications, 37(2010) 3352-3357.

[10] WoodA.J.,WollenbergB.F., Power generation operation and control, John wiley& Sons, 1996.[11] YanH.,LuhP.B., GuanX., RoganP.M., Scheduling of hydrothermal power systems, IEEE transactions on

power systems, 8(1993) 1358-1365.[12] ZaghloolM.F., TruttF.C., Efficient methods for optimal scheduling of fixed head hydrothermal powersystems, IEEE transactions on power systems,3 (1988) 24-30.

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8/9/2019 IJM2C - Vol 4, No 3, SN 15

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A. Hussain, A. Qayyuma/, 04-03 (2014) 223-241

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processing, CO2 capture from flue gases, nitrogen production from air, separation of

hydrogen streams in refinery and petrochemical process as well as in ammonia and

methanol process streams [1-3]. The feasibility of membrane ba

ijm2c - vol 4, no 3, sn 15

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