Economic Dispatch Scheduling using Classical and Newton Raphson Method

Transcription

1 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 Volume, Issue, June0 Internatonal Journal of Enneerng and Management Research age umber: Economc Dspatch Schedulng usng Classcal and ewton Raphson Method avneet Kaur, Mannder, Inderjeet Sngh,, Electrcal Enneerng & TU, IDIA ABSTRACT Over the past several years, concerns have been rased over the possblty that the exposure to 00 Hz electromagnetc felds from power lnes, substatons and other power sources may have detrmental health effects on lvng organsms The economc dspatch problem was defned so as to determne the allocaton of electrcty demand among the commtted generatng unts to mnmze the operatng costs subject to physcal and technolocal constrants Economc load dspatch s an mportant optmzaton task n power system operaton for allocatng generaton among the commtted unts such that the system constrants mposed are satsfed and energy requrement n terms of kcal/h or Btu/h or Rupees per hour (Rs/h) are mnmzed To all ntents and purposes, there has been concern that the economc dspatch may not be the best envronmentally In ths research paper economc schedulng of thermal unts has been done Over and above, regular electrc supply s the sheer necessty for growng ndustry and other felds of lfe Keywords Economc dspatch, amda teraton, Classcal Method and ewton Raphson Method I ITRODUCTIO The economc load dspatch problem pertans to the optmum generaton schedulng of avalable generatng unts n a power system to mnmze the cost of generaton subject to system constrants [,] In vew of rapd growth n demand and supply of electrcty, electrc power system s becomng ncreasngly larger and more complex day by day Regular electrc supply s the utmost necessty for growng ndustry and other felds of lfe Wth the ncreasng dependence of ndustry, agrculture and daytoday household comfort upon the contnuty of electrc supply, the relablty of power systems has put on great mportance [] Every electrc utlty s normally under oblgaton to provde to ts consumers a certan degree of contnuty and qualty of servce (power flow on transmsson lnes n a specfed range) Therefore, economy, emsson etc objectves of the power system must be properly coordnated n arrvng at optmal power dspatch[,] It s, therefore, requred to search for better and realstc stratees to acheve varous objectves along wth desred qualty of power supply and satsfyng smultaneously varous system constrants Ths mples economc load dspatch schedulng aspects of the system operaton, whch are duly nvestgated n the present work n a unfed multobjectve approach[] II ECOOMIC DISATCH Economc dspatch n electrc power system has ganed ncreasng mportance as the cost assocated wth generaton and transmsson of electrc energy keeps ncreasng So n optmal load dspatch problem generator operatng cost characterstcs s the most mportant factor The major component of generaton operatng cost for fossl plants s the cost of fuel nput per hour, whle the cost of mantenance, water etc contrbute only neglbly small portons[] The operatng characterstcs of fossl plants can be expressed n terms of mllon calores per uneconomcal (or may be techncally nfeasble) to operate the unt and MW Copyrght 0 Vandana ublcatons All Rghts Reserved

2 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 lmts [,9] Mathematcally, ths s an optmzaton problem III SOUTIO BASED O AMBDA ITERATIO METHOD The economc dspatch problem s defned as that whch mnmzes the total operatng cost of the power system whle meetng the total load plus transmsson losses wthn generator lmts Mathematcally, the problem s defned a Fg Operatng cost versus power output curve The power output of the plant s ncreased sequentally by openng a set of valves at the nlet to ts steam turbne The throttlng losses n a valve are large, when t s just opened and small when t s fully opened As a result the operatng cost of a plant has the form as shown n Fgure MW (mn) MW (max) ( Y axs Operatng cost Rs/h ) ower Output Fgure Cost Rs/h versus output MW curve For dspatchng purposes, ths operatng cost per unt generator s usually approxmated by quadratc polynomal ( ) ( F = a + b + c ) g g g Rs /h Where a, b, c are the cost coeffcents and stands for the unt s number The man objectve of the economc dspatch s to mnmze the cost of fuel for the thermal power system, subject to certan constrants as generatng enough power to meet the load demand and to stay wthn operatng Mnmze F ( ) = ( a + b + c ) = (a) Subject to () The energy balance equaton = = D + (b) Rs/h () and the nequalty constrant mn max ; ( =, ) = = j=,, (c) Where a,b, c are the cost coeffcents D s the load demand s the real power generaton and wll act as decson varable s the number of generaton buses s the transmsson power loss One of the most mportant, smple but approxmate method of expressng transmsson loss as a functon of generator powers s through Bcoeffcents Ths method uses the fact that under normal operatng condtons, the transmsson loss s quadratc n the njected bus real powers The general form of the loss formula usng Bcoeffcents s B j gj MW () Where, and gj are the real power njectons at the th and jth buses, respectvely B j are the loss coeffcents whch are constant under certan assumed condtons, s number of generaton buses The transmsson loss formula of () s known as George s formula Another more accurate form of transmsson loss expresson, frequently known as Kron s loss formula s Copyrght 0 Vandana ublcatons All Rghts Reserved

3 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 = B 00 + = B 0 + = j= MW () Where and are the real power njectons at th gj and jth buses, respectvely B 00, B 0 and B j are the loss coeffcents whch are constant under assumed condtons, s the number of the generaton buses The above constraned optmzaton problem s converted nto an unconstraned optmzaton problem agrange multpler method s used n whch the functon s mnmzed (or maxmzed) wth sde condtons n the form of equalty constrants Usng agrange multplers and augmented functon s defned as (, λ) = F ( ) + λ ( D + ) () = Where λ s the agranan multpler ecessary condtons for the optmzaton problem are (, λ )/ = F( )/ + λ ( / ) =0 (=,, ) Rearranng the above equaton, F( )/ = λ ( / ) (=,, ) () Where F( )/ s the ncremental cost of the th generator (Rs/MWh) / s the ncremental transmsson losses Equaton () s known as the exact coordnaton equaton, and (, λ ) λ / = D + = 0 () = Equaton () the socalled coordnaton equaton, numberng s solved smultaneously wth Eqn () to yeld a soluton for agrange multpler λ and the optmal generaton of generators, By dfferentatng the transmsson loss equaton Eqn () wth respect to, the ncremental transmsson loss can be obtaned as, / = B 0 + Bjgj (=,, ) () j= And by dfferentatng cost functon eqn(a) wth respect to, the ncremental cost can be obtaned as B j gj F()/ = a +b ( ) Equaton () can be rewrtten as (=,, ) { F( )/ } / ( / )= λ or { F( )/ } = λ (=,, ) (9) where = 0/( / ) s called the penalty factor of th plant To obtan the soluton, substtute Eqs () and () nto eq () a +b = λ (B 0 B j gj ) (=,, ) j= Rearranng the above equaton to get, we have (a + λ B ) =λ (B 0 B j gj ) b (=,, ) j= j The value of can obtaned as = {λ (B 0 B j gj )b} / (a +λ B ) (=,, ) (0) j= j If the ntal values of (=,, ) and λ are known the above equaton can be solved teratvely untl eq() s satsfed by modfyng λ Ths technque s known as successve approxmaton IV AGORITHM: ECOOMIC DISATCH (CASSICA METHOD) Read data, namely cost coeffcents, a, b, c ; B coeffcents, B j, B 0, B 00 (=,, ; j=,, ); convergence tolerance є, step sze α, and maxmum teratons allowed, ITMAX,etc Compute the ntal values of (=,, ) and λ by assumng that the transmsson losses are zero, e =0 Set teraton counter, IT = Compute (=, ) usng eq(0) Compute transmsson loss usng Eq () Compute = D + = Check ε,f yes, then GOTO Step 0 Check IT ITMAX, f yes then GOTO Step 0(t means program termnated wthout obtanng the requred convergence) Copyrght 0 Vandana ublcatons All Rghts Reserved

4 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 Update λ new =λ+α, where α s the step sze used to ncrease or decrease the value of λ n order to meet the step 9 IT =IT+, λ= λ new and GOTO Step and repeat 0 Compute optmal total cost from eq(a) and transmsson loss from () Stop V ECOOMIC DISATCH USIG EWTORAHSO METHOD The economc dspatch problem s expressed by eqs(a), (b) and (c) and s converted nto an unconstraned optmzaton problem as n eq() ecessary condtons for the optmzaton problems eq() are ven by eqs () and eqs () The soluton of nonlnear eq() can be obtaned usng the ewton Raphson method n whch any change n control varables about the ntal values can by obtaned usng Taylor s expanson Taylor s expanson to second order of eq () and eq () can be wrtten as ( / ) + ( / gj ) gj + ( / λ) λ = / () j= j (( /( λ gj ) gj + ( / λ ) λ = / λ () j= The above equaton can be rewrtten n matrx form as p g p g p g λ p g = p g T λ p g λλ λ λ () Dervatves can be obtaned as follows: / = F/ + λ [( / ) ] =( a +b ) + λ (B 0 + B j gj ) = (=,, ) (a) / λ = D + (b) = Takng dervatves of eq(a) wth respect to, ( / ) = F/ + λ( / ) = a +λ B (=,, G) ( a) ( / gj ) = λ ( / ) = λ B j (b) (=, G; j=,, G; j) Takng dervatves of eqs (a) and ( b) wth respect to λ, G ( / λ )= ( / λ) = / =B 0 + B j gj = (=,, G) (c) / λ =0 (d) Equatons () and () are terated tll no further mprovement s obtaned or sngle dervatves wth respect to control varables become zero VI AGORITHM: ECOOMIC DISATCH (EWTORAHSO METHOD) Read data, namely a, b, c (cost coeffcents); B j, B 0, B 00 (Bcoeffcents) (=,, ; j=,, ) convergence tolerance, є, and ITMAX ( maxmum allowed teratons), etc Compute the ntal values of (=,,,) and λ by presumng that = 0 Assume that no generator has been fxed ether at lower lmt or at upper lmt Set teraton counter IT = Compute Hessan and Jacoban matrx elements usng eqs () and () [H] p g = [J] λ Deactvate row and column of Hessan matrx and row of Jacoban matrx representng the generator whose generaton s fxed ether at lower lmt or at upper lmt Ths s done so that fxed generators,can not partcpate n allocaton Gauss elmnaton method s employed n whch trangularzaton and backsubsttuton processes are performed to fnd (=,, R and λ) Here R s the number of generators whch can partcpate n allocaton or Check ether R = R = + λ ( ) + ( λ) ε ε Copyrght 0 Vandana ublcatons All Rghts Reserved

5 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 f convergence condton s yes then GOTO Step 0 Check IT > ITMAX, f condton s yes GOTO Step 0(t means the procedure proceeds wthout obtanng requred convergence) Modfy control varables, (new) = + ; (=,, R ) and λ (new) = λ + λ 9 IT =IT+, = (new), λ = λ (new) and GOTO Step and repeat 0 If no more volatons then GOTO Step, else check the lmts of generators and fx up as follows: If < mn then = mn If > max then = max GOTO Step and repeat Compute the optmal total cost and transmsson loss Stop VII TEST SYSTEMS AD RESUTS Test problem no The fuel nputs per hour of two plants are ven as F (g ) = (0009 g +0 g +00) Rs/h F (g ) = (000 g +0 g +0) Rs/h Determne the economc schedule to meet the demand of 0MW and the correspondng cost of generaton The transmsson losses are ven by = 000g g (0000 g g ) Assumptons: α = 00, ε =0000, and ITMAX= TOTA COST = 09 Rs/hr Table Generaton Schedule (R Method) IT g (M g (M λ(rs/mwh) OWER OSS = 99 MW TOTA COST = 09 Rs/hr Test problem no For a three generator system the fuel cost coeffcents are ven n table (a) The B coeffcents for transmsson loss are ven n table (b)determne the economc schedule for load of 0 MW Table Generaton Schedule (Classcal Method) I T g (M g (M λ(rs/mw h) Δ(M (M Table (a) Fuel cost coeffcents Table (b) Bcoeffcents MW Table Optmal Generaton Schedule (R Method) IT g (M g (M g (M λ(rs/mwh) Copyrght 0 Vandana ublcatons All Rghts Reserved

6 wwwjemrnet ISS (OIE): 00, ISS (RIT): 99 OWER OSS = 09 MW TOTA COST = Rs/hr Table Generaton Schedule (Classcal Method) I T g ( M g ( M g ( M λ(rs/m Wh) Δ(M (M TOTA COST = Rs/hr VIII COCUSIO Economc dspatch s the shortterm determnaton of the optmal output of a number of electrcty generaton facltes, to meet the system load, at the lowest possble cost, subject to transmsson and operatonal constrants The basc constrants of the economc dspatch problem reman n place but the model s optmzed to mnmze pollutant emsson n addton to mnmzng fuel costs and total power loss Due to the added complexty, a number of algorthms have been employed to optmze ths envronmental/economc dspatch problem It s concluded that the economc schedulng of thermal unts to meet the load demand n the most economc way wthout volatng any system or ndvdual unt constrants REFERECES [] BHChowdhary and SRahman, A revew of recent advances n economc dspatch, IEEE Trans on ower Systems, Vol, no, pp,990 [] MHuneault and FDGalana, A survey of the power flow lterature, IEEE Trans on ower Systems, Vol, no, pp0, 99 [] JSDhllon, SCart and DKothar, Stochastc economc emsson load dspatch, Electrc ower Systems Research, vol, pp 9, 99 [] Janda, D Kothar and KS ngamurthy, Economc emsson load dspatch through goal programmng technques, IEEE Transactons on Energy Converson, vol EC (), pp, 9 [] C Durga rasad, AK Jana and SC Trpathy, Modfcatons to ewton Raphson load flow for llcondtoned power systems, Electrcal ower & Energy Systems, vol (), pp 99, 990 [] AO Ekwue and JF Macqueen, Comparson of load flow soluton methods, Electrc ower Systems Research, vol (), pp, 99 [] C alanchamy & K Shrkrshna, Smple Algorthm for economc power dspatch, vol, 99 [] HW Dommel & WF Tnney, Optmal power flow solutons, IEEE Trans, Vol AS, 9 [9] K Aok & T Satoh, ew Algorthms for classc economc load dspatch", IEEE Trans Vol 0, 9 Copyrght 0 Vandana ublcatons All Rghts Reserved