IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 7, Issue 1 (Jul. - Aug. 2013), PP 55-59 www.iosrjournals.org

An Application Jeevan â&#x20AC;&#x201C; Kushalaiah Method to Find Lagrangian Multiplier in Economic Load Dispatch Including Losses and Lossless Transmission Line Neelam Jeevan Kumar Electric and Electronics Engineering, H.No: 19-6-194, Rangashaipet, Warangal, Andhra Pradesh, India506005

Abstract: This paper discusses the non iterative method to calculate the Lagrangian Multiplier ( đ?&#x153;&#x2020;). The economic load dispatch (ELD) is the on-line economic load dispatch (ELD) wherein it is required to distribute the load among the generating units. The system such manner as to minimize the total cost of supplying the minute -to- minute requirements of systems. Jeevan â&#x20AC;&#x201C; Kushalaiah method or J-K method is a method to calculate maximum number of combination between n-elements starting from minimum order to maximum order. The J-K method modifies iterative flow charts to single line flow charts. This paper deals with NewtonRaphson method. Index terms: Jeevan â&#x20AC;&#x201C; Kushalaiah method, Incremental Fuel Rate, Incremental Transmission Cost

I.

Introduction

In load flow studies, For a particular load demand the generation at all generator buses are fixed except at the generator bus known as slack bus or reference bus or swing bus. In case of Economic Load Dispatch (ELD) the generations are not fixed but they are allowed to take values again the within certain limits so as to meet a particular load demand with minimum fuel consumption. The cost of generation will depend upon the system constraint for a particular load demand. This means the cost of the generation is not fixed for a particular load demand but depends upon the operating constrains of the generator. The various constraints in economic load dispatch are Equality Constrains: Pp-jQp = Vp*Ip = Vp* nq=1 YpqVq â&#x20AC;Śâ&#x20AC;Ś (i) Where p = intial bus and q = terminating bus Vp = ep+jfp and Ypq = Gpq -jBpq Pp = đ?&#x2018;&#x203A;đ?&#x2018;&#x17E;=1{ep(eqGpq+fqBpq)+fp(fqGpq-eqBpq)} â&#x20AC;Śâ&#x20AC;Ś (ii) Qp = đ?&#x2018;&#x203A;đ?&#x2018;&#x17E;=1{fp(eqGpq+fqBpq)+ep(fqGpq-eqBpq)} â&#x20AC;Śâ&#x20AC;Ś (iii) The limits of p and q: 1 â&#x2030;¤ p â&#x2030;¤ n and 1 â&#x2030;¤ q â&#x2030;¤ n

Y-axis (Fuel input in million Btu/hr)

Inequality Constrains: a) Generator Constraints: Pp2 + Qp2 â&#x2030;¤ Cp2, where Cp is prespecified value. Ppmin â&#x2030;¤ Pp â&#x2030;¤ Ppmax and Qpmin â&#x2030;¤ Qp â&#x2030;¤ Qpmax b) Voltage Constraints: đ?&#x2018;&#x2030;đ?&#x2018;?đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2030;¤ đ?&#x2018;&#x2030;đ?&#x2018;? â&#x2030;¤ đ?&#x2018;&#x2030;đ?&#x2018;?đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ áľ&#x;pminâ&#x2030;¤áľ&#x;â&#x2030;¤áľ&#x;pmax Network Security Constraints: Incremental Fuel rate =

(â&#x2C6;&#x2020; input )

(â&#x2C6;&#x2020; output )

==

đ?&#x2019;&#x2026;(đ??˘đ??§đ??Šđ??Žđ??­) đ?&#x2019;&#x2026;(đ??¨đ??Žđ??­đ??Šđ??Žđ??­)

=

đ?&#x2018;&#x2018;đ??š đ?&#x2018;&#x2018;đ?&#x2018;&#x192;

X-axis(Power ouput in MW)

in Rs/Btu

Incremental Efficiency = reciprocal of Incremental fuel Rate =

II.

đ?&#x2019;&#x2026;đ?&#x2018;ˇ đ?&#x2019;&#x2026;đ?&#x2018;­

Fig.1: Incremental fuel rate vs. power output

Jeevan â&#x20AC;&#x201C; Kushalaiah Method

Jeevan â&#x20AC;&#x201C; Kushalaiah method is a method to find maximum number of possible combination between nelements. Let n-elements a1,a2,a3,a4, â&#x20AC;Ś an-1, an The maximum number of combination between elements are www.iosrjournals.org

55 | Page

An Application Jeevan â&#x20AC;&#x201C; Kushalaiah Method To Find Lagrangian Multiplier In Economic Load {(1),( a1,a2,a3,a4, â&#x20AC;Ś an-1, an),( a1a2, a1a3, a1a4, â&#x20AC;Ś a1an-1, a1an.. a2a3, a2a4, â&#x20AC;Ś a2an-1, a2anâ&#x20AC;Śâ&#x20AC;Ś a3a4, a3a5, â&#x20AC;Ś a3an-1, a3anâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. an-1an),( a1a2a3, a1a2a4, â&#x20AC;Ś a1a2an-1, a1a2anâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śan-2an-1an),â&#x20AC;Ś..,(â&#x20AC;Ś..),â&#x20AC;Śâ&#x20AC;Ś.,(a1a2a3â&#x20AC;Śan1an)} â&#x20AC;Ś.. (iv) = {(Ď´0),( Ď´1),( Ď´2),â&#x20AC;Śâ&#x20AC;Ś.( Ď´p-1),( Ď´p)} Here Ď´0 = 1, Ď´1 = +,-,*,/ of all elements ( Ď´p) = Addition or Subtraction or Multiplication or Division or a square Matrix. Example - 1: ( Ď´p) = (a,b) ď&#x201A;ˇ Addition = a + b ď&#x201A;ˇ Subtraction = a - b ď&#x201A;ˇ Multiplication = a*b ď&#x201A;ˇ Division = a/b đ?&#x2018;&#x17D;11 12 ď&#x201A;ˇ Square Matrix = (a and b are diagonal positions only.) 21 đ?&#x2018;?22 Commonly Multiplication and Addition is performed Ď´p = (n - (p-1)) p - 1 â&#x20AC;Śâ&#x20AC;Ś.. total number of combinations of n-elements = Ď´0 +Ď´1 + 2đ?&#x2018;?=2 Ď´p â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. The value of p â&#x2030;Ľ 2 where

(v) (vi)

is summator like factorial operation instead of multiplication addition is performed.

Example-2: (5)2

=5

= 5 +4 +3 +2 +1

III.

= 5+4+3+2+1+4+3+2+1+3+2+1+2+1+1 = 35

Calculation of Îť of ELD without Losses

The Load demand = PD The generation of n th unit = Pn Total fuel input to the system = FT The Fuel input tot nth system = Fn The system with without losses,PL = 0 PD = nk=1 Pk â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (vii) The auxiliary function F = FT+Îť(PD - nk=1 Pk) Where Îť is Lagrangian multiplier Differentiating F with respect to Pn and equating to zero đ???đ??&#x2026;đ?&#x;?

dFn

đ???đ???đ?&#x;?

=

đ???đ??&#x2026;đ?&#x;?

đ???đ???đ?&#x;?

=

đ???đ??&#x2026;đ??§

đ???đ???đ??§

=Îť

Where = incremental production cost of nth plant in dPn Rs. /MWhr For a plant over a limited range đ???đ??&#x2026;đ??§ đ???đ???đ??§

= FnnPn +fn = Îť

Fig .2: Generalized diagram of a power system without losses.

â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (viii)

Where Fnn = slope of total incremental production curve and fn = intercept of incremental production cost curve. From equation number (viii), we have Pn = (Îťassum â&#x20AC;&#x201C; fn) / Fnn P1 = (Îťassum â&#x20AC;&#x201C; f1) / F11 , F2 = (Îťassum â&#x20AC;&#x201C; f2) / F22â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś Pn-1 = (Îťassum â&#x20AC;&#x201C; fn-1) / Fn-1n-1 ,Pn = (Îťassum â&#x20AC;&#x201C; fn) / Fnn Total power to be generated PT is PD = P1 + P2 +â&#x20AC;Ś..+ Pn www.iosrjournals.org

56 | Page

An Application Jeevan â&#x20AC;&#x201C; Kushalaiah Method To Find Lagrangian Multiplier In Economic Load PD = {(Îťassum â&#x20AC;&#x201C; f1) / F11 }+{(Îťassum â&#x20AC;&#x201C; f2) / F22 }+â&#x20AC;Ś+{(Îťassum â&#x20AC;&#x201C; fn) / Fnn } â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (ix) Rewriting equation-(ix) for direct calculation of Îť

Îťassum =

đ??§ đ??Ł=đ?&#x;? đ??&#x2026;jj)

{PD (

+

đ??§ đ??¤=đ?&#x;? đ??&#x;k(Ď&#x2022;n-k-1)}/(

đ??§ đ??§=đ?&#x;? đ?&#x203A;&#x; đ??§)

â&#x20AC;Śâ&#x20AC;Ś

(x)

Where Ď&#x2022;n = elements combination in Ď´n-1 IV.

đ?&#x153;&#x2022;đ?&#x2018;&#x192;L

Calculation of Îť of ELD without Losses

The system with without losses,PL PD + PL = đ??§đ??¤=đ?&#x;? đ???k â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (xi) The auxiliary function F = FT + Îť ( PD + PL - đ??§đ??¤=đ?&#x;? đ???k ) đ???đ??&#x2026;đ??§ đ?&#x203A;&#x203A;đ???đ??&#x2039; + đ?&#x203A;&#x152;(đ?&#x203A;&#x203A;đ???đ??§) = đ?&#x203A;&#x152; â&#x20AC;Ś.. (xii) đ???đ???đ??§

= The Incremental Transmission Loss at plant â&#x20AC;&#x201C; n Loss formula approximately PL = đ??Ł đ???j đ??¤ đ???k Bjk Assumptions ď&#x201A;ˇ The equivalent load current at any bus remains constant. ď&#x201A;ˇ The generator bus voltage magnitude and angles are constant. ď&#x201A;ˇ Power factor is constant đ?&#x203A;&#x203A;đ???đ??&#x2039; = 2 đ?&#x2019;&#x2039; đ?&#x2018;ˇjBjk â&#x20AC;Ś.. (xiii) đ?&#x203A;&#x203A;đ???đ??§ đ?&#x153;&#x2022;đ?&#x2018;&#x192;đ?&#x2018;&#x203A;

đ?&#x2019;&#x2026;đ?&#x2018;­đ?&#x2019;?

đ?&#x2019;&#x2026;đ?&#x2018;ˇđ?&#x2019;?

= FnnPn + fn

â&#x20AC;Ś..

(xiv)

Fig . 3: Generalized diagram of a power system with losses.

Substituting equations (xiii)(xiv) in (xii) we get Rewriting the equation - (xii)

FnnPn+fn+2Îť đ?? mnPm = Îť đ???đ??&#x2026;đ??§ đ???đ???đ??§

đ???đ??&#x2026;đ??§ đ???đ???đ??§

đ?&#x203A;&#x203A;đ???đ??&#x2039;

â&#x20AC;Śâ&#x20AC;Ś

(xv)

đ?&#x203A;&#x203A;đ???đ??&#x2039;

+ đ?&#x203A;&#x152;(đ?&#x203A;&#x203A;đ???đ??§) = đ?&#x203A;&#x152; , đ?&#x203A;&#x203A;đ???đ??§ = Ln đ???đ??&#x2026;đ??§

+ ÎťLn = Îť

đ???đ???đ??§

From equation â&#x20AC;&#x201C; (xiv) in (xvi)

= Îť (1- Ln) â&#x20AC;Śâ&#x20AC;Ś

(xvi)

FnnPn + fn = Îť (1- Ln) (FnnPn + fn)/( 1- Ln) = Îť Generation of n-generator Pn = {Îť (1- Ln) â&#x20AC;&#x201C; fn}/Fnn P1 = {Îť (1- L1) â&#x20AC;&#x201C; f1}/F11 , P2 = {Îť (1- L2) â&#x20AC;&#x201C; f2}/F22 â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś Pn = {Îť (1- Ln) â&#x20AC;&#x201C; fn}/Fnn PT = P1 +P2 + Pn = {Îť (1- L1) â&#x20AC;&#x201C; f1}/F11 + {Îť (1- L2) â&#x20AC;&#x201C; f2}/F22 +â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś + {Îť (1- Ln) â&#x20AC;&#x201C; fn}/Fnn Rewriting the above equation

Îťassum = {PD (

đ??§ đ??Ł=đ?&#x;? đ??&#x2026;jj)

V.

+

đ??§ đ??¤=đ?&#x;? đ??&#x;k(Ď&#x2022;n-k-1)}

/(

đ??§ đ??§=đ?&#x;? đ?&#x203A;&#x; đ??§)(1-Ln)

â&#x20AC;Śâ&#x20AC;Ś.. ( xvii )

Algorithms and Flow Charts

V.a. Algorithm for ELD without Losses and ELD with losses Step-I. Start and Read The Fuel input tot n th system ( Fn ), Fnn = slope of total incremental production, The Load demand ( PD ) Step-II. Calculate loss. For ELD without losses PL = 0 Step-III. Calculate Îťassum, and set bus number, n = 1, calculate Pn check all buses are completed or not, if yes go to next step, if not increment n = n+1 repeat step-III Step-IV Print Generation and calculate cost of generation. Note: Algorithm for ELD without Losses and ELD with losses changes according to their transmission line losses and operating conditions. In these algorithms are only for normal operation and not affected for atmospheric changes like raining and snow falling on the line. www.iosrjournals.org

57 | Page

An Application Jeevan â&#x20AC;&#x201C; Kushalaiah Method To Find Lagrangian Multiplier In Economic Load Flow chart for ELD without losses

Flow chart for ELD with losses

www.iosrjournals.org

58 | Page

An Application Jeevan â&#x20AC;&#x201C; Kushalaiah Method To Find Lagrangian Multiplier In Economic Load V.

Conclusion

In this method, the Calculated values are within the limits of voltage constraints and generator i.e.,

đ?&#x2018;&#x2030;đ?&#x2018;?đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2030;¤ đ?&#x2018;&#x2030;đ?&#x2018;? â&#x2030;¤ đ?&#x2018;&#x2030;đ?&#x2018;?đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ and Ppmin â&#x2030;¤ Pp â&#x2030;¤ Ppmax and Qpmin â&#x2030;¤ Qp â&#x2030;¤ Qpmax And there is iteration process to calculate Lagrangian multiplier Îť,in the flow chart backward path is only for calculating required output at generating stations not to finding out the Îť. This method applied for Thermal and Hydro power stations or connection of Hydro-Thermal Power station and Grid connections.

References [1]. [2]. [3]. [4]. [5]. [6]. [7].

Neelam Jeevan Kumar,â&#x20AC;? Jeevan kushalaiah method to find the coefficients of characteristic equation of a matrix and introduction of summetorâ&#x20AC;?, 2229-5518, Volume 4, Issue 8,2013 I.A . Farhat , M.E. El-Hawar y, â&#x20AC;&#x153;Optimization methods applie d for solving the short-term hydrothermal coordination problemâ&#x20AC;?, 79 (20 09) 1308â&#x20AC;&#x201C;1320, doi:10.1016/j.epsr.20 09.0 4.0 01 Y.H. Song, Modern Optimization Techniques in Power Systems, Kluwer Aca-demic Publishers, Dordrecht, The Netherlands, 1999 J.A . Momoh, Electric Power System Applications of Optimization, Marcel Dekker, New York , 20 01. Basic issues in Lagrangian optimization, in Optimization in Planning and Operation of Electric Power Systems (K. Frauendorfer, H. Glavitsch and R. Bacher, eds.), Physica-Verlag, Heidelberg, 1993, 3-30 (by R. T. Rockafellar) R. T. Rockafellar â&#x20AC;&#x153;Lagrange Multipliers and Optimalityâ&#x20AC;?, 35(2), 183â&#x20AC;&#x201C;238. (56 pages), http://dx.doi.org/10.1137/1035044 book: C.L Wadhwa, â&#x20AC;?Electrical power systemsâ&#x20AC;?, ISBN 978-81-224-2839-1

Author The author is academic researcher, Department Electrical and Electronics Engineering. His research areas are Optimization of Power system and applied mathematics.

www.iosrjournals.org

59 | Page

An Application Jeevan – Kushalaiah Method to Find Lagrangian Multiplier in Economic Load Dispatch I

http://www.iosrjournals.org/iosr-jeee/Papers/Vol7-issue1/I0715559.pdf?id=5802