Yugoslav Journal of Operatons Research (00) umber 57-66 A SEPARABLE APPROXIMATIO DYAMIC PROGRAMMIG ALGORITHM FOR ECOOMIC DISPATCH WITH TRASMISSIO LOSSES Perre HASE enad MLADEOVI] GERAD and Ecole des Hautes Etudes Commercales Montr al (Qu bec) Canada Abstract: The standard way to solve the statc economc dspatch problem wth transmsson losses s the penalty factor method. The problem s solved teratvely by a Lagrange multpler method or by dynamc programmng usng values obtaned at one teraton to compute penalty factors for the next untl stablty s attaned. A new teratve method s proposed for the case where transmsson losses are represented by a quadratc formula (.e. by the tradtonal B-coeffcents). A separable approxmaton s made at each teraton whch s much closer to the ntal problem than the penalty factor approxmaton. Consequently lower cost solutons may be obtaned n some cases and convergence s faster. Keywords: Economc dspatch transmsson losses B-coeffcents penalty factors separable approxmaton dynamc programmng.. ITRODUCTIO Due to the enormous costs nvolved optmzng the use of equpment for power generaton and transmsson s a lastng concern. Gven several thermal or hydro unts often wth dfferent characterstcs one must decde how to dstrbute the load consdered between them. Ths problem called economc dspatch has been much studed both n the statc and dynamc cases. It s dscussed at length n the book of Wood and Wollenberg [6] Power generaton operaton and control an enlarged second edton of whch has recently appeared. The standard way to solve the statc case wth transmsson losses s the penalty factor method. The problem s solved teratvely by a Lagrangan multpler method or by dynamc programmng usng values obtaned at one teraton to compute penalty factors for the next untl stablty s attaned.
58 P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm The purpose of ths paper s to propose a new teratve method for the case where transmsson losses are represented by a quadratc formula (.e. by the tradtonal B-coeffcents). A separable approxmaton s made at each teraton whch s much closer to the ntal problem than the penalty factor approxmaton. Moreover the problem consdered at each teraton can be solved by dynamc programmng. Convergence s faster than n the scheme of to Lang and Glover [4] whch also uses dynamc programmng. The paper s organzed as follows: a mathematcal formulaton of the problem s gven n the next secton. Prevous soluton methods are revewed n Secton 3. The partcular case of a separable loss functon s studed n Secton 4 and llustrated by an example of Wood and Wollenberg [6]. The new method s descrbed n full n Secton 5 and appled to examples from [6] and [4]. Conclusons are drawn n Secton 6.. FORMULATIO Consder thermal unts commtted to serve a load of P R at mnmum cost. Assume that producton of each unt s bounded below and above. Assume further that transmsson losses n the network are ncurred and can be represented by a quadratc formula (wth the so-called B-coeffcents). Ths problem can be stated mathematcally as mnmze FT = F( P ) = () subject to and P = PR + P L () = Pmn P Pmax =... (3) where P s the output of unt (n MW) F( P) s the nput of unt or ts cost rate (n $/h ) P R s the requred load (n MW) P L s the transmsson losses (n MW). Moreover PL = PB jpj + B0P0 + B 00. (4) = j= = Dervaton of ths formula for losses s explaned n [6]. eglectng transmsson losses.e. settng P L = 0 n () problem () - (3) can be solved by a Lagrange multpler method [6] or by dynamc programmng [ 4 6].
P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 59 3. PREVIOUS SOLUTIO METHODS When transmsson losses are consdered the standard soluton method s the penalty factor approach. Consder the Lagrange functon wth a multpler λ for (): mnmze L = F( P) + λ( PR + PL P). (5) = = eglectng bounds on the where P the frst order condtons are equaton () above and df = λ =... (6) L L = B P + B0 j= j j (7) are the ncremental losses and = L are the penalty factors coordnaton equatons. If bounds on the BjPj B0 j= = PF (8) PF for unts =.... Equatons () and (6) are called P are taken nto account the frst order condtons become λ f P = Pmn df PF = λ f P ( Pmn P max) λ f P = Pmax. (9) An teratve method to solve ()-(3) s the followng [6]:. Solve the coordnaton equatons wth unt penalty factors (.e. neglectng losses) to 0 0 0 get an ntal soluton P P... P. Set the teraton counter k =.. Compute penalty factors wth values of the last teraton.e. PF ( k) = =... n B P j= j ( k ) j B 3. Solve the coordnaton equatons to get a soluton 0 ( k) ( k) ( k)... P P P. (Ths can be done n varous ways e.g. a search technque for the optmal λ a ewton-raphson search etc see [6]).
60 P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 4. Compute the current dfference between the producton and load plus losses.e. ( k) ( k) ( k) ( k) P PR P BjPj B0P B 00 = = j= = δ = If δ ε (a gven tolerance) stop. Otherwse ncrease k by and return to step. Whle no formal proof of convergence for ths method seems to have been publshed and values of δ may oscllate a local and possbly global optmum s usually reached n a farly small number of teratons. Modfcaton to the teratve method just descrbed to allow soluton by dynamc programmng are proposed n [4]. In Step the ntal soluton s found by the recurson where * * F ( D ) = mn{ F ( ) + ( )} D P R P F P (0) R s the set of ntegers n [ P P mn max] the range of producton for unt assumng that approxmaton to MW and D s chosen among the range of possble productons for the frst unts.e. t s nteger and kmn kmax k= k= P D P. () k In Step 3 each cost functon F s multpled by the penalty factor P the load P R s augmented by the losses as estmated and the problem s agan solved by the dynamc programmng recurson (0). The Lagrange functon for ths step s F( P) = ( k ) BjPj B0 j= ( k ) ( k ) ( k ) + λ P + + + R P BjPj B0 P B00 P = j= = = () and the frst order condton as the losses are fxed = λ f P = P BjP B f P = P j= mn df = λ f P ( Pmn Pmax ( k ) j 0 λ max. and are the same as (9) except that the values of ) (3) P are fxed n the penalty factors.
P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 6 4. SEPARABLE LOSSES Economc dspatch wth transmsson losses can be optmzed by dynamc programmng wthout usng penalty factors. The easest case s when losses are separable n the P. Assume also that they are quadratc a partcular case consdered n [6]. Then the power load and the losses. Thus P produced by unt wll be equal to the power P gong to the P = P b0 bp bp (4) where b b and are the known coeffcents of the loss functon. Hence 0 b b ( b) 4b( P + b P = 0 ). (5) b One can assocate to P a cost functon F ( P ) equal to the cost necessary for unt to contrbute P to the load.e. F( P) where P s obtaned from P by (4). The correspondence between F and F s llustrated on Fgure. Fgure : Input-Output functons F( P ) and F( P ) The range of values of P deduced from that of values or n other words an approxmaton of MW: P s assumng nteger Pmn = [ Pmn b0 bpmn bpmn ] P [ Pmax b0 bpmax bpmax ] = P max (6)
6 P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm where a and a are respectvely the smallest nteger not smaller than and the largest nteger not larger than a ( P mn and P max are assumed to be nteger as well). The dynamc programmng recurson s * * F ( D ) = mn{ F ( ) + ( )} D P R P F P (7) R = [ P P ] the range of effectve producton of unt (.e. that whch s where mn max not lost) and D Pk mn P k max (8) k= k= the range of demand whch can be satsfed by the frst unts. Ths recurson s used * * to fnd the optmal polcy... * P P P from whch the powers P * P *... P * to be produced by each unt are obtaned by (5). The cost of ths polcy s * * F ( PR) = ( = F P ). Observe that the method proposed gves a globally optmal soluton (up to the approxmatons n the values of the P ) even f the cost functons F( P) and/or P are non-convex. In contrast the classcal penalty functon method may stop at a local optmum n ths case. Example. (Wood and Wollenberg [6] p 36). Ths problem nvolves three unts wth nput/output functons and producton ranges: PR F ( P ) = 56. 0 + 7. 90P + 0. 0056P P = 50 MW P = 600 MW mn max F ( P ) = 30. 0 + 7. 850P + 0. 00940P P = 00 MW P = 400 MW mn max 3 3 3 3 F ( P ) = 78. 0 + 7. 970P + 0. 0048P P = 50 MW P = 00 MW 3mn max = 850 MW and losses are gven by PL = 0. 00003P + 0. 00009P + 0. 000P 3. Soluton. The followng soluton s obtaned by applyng dynamc programmng to the transformed problem: P = 434. 668 P = 300. 06 P = 3. 06 wth losses equal to 3 5.8350. ote that the same losses are obtaned by the penalty factor approach n 4 teratons ([6]).
P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 63 5. SEPARABLE QUADRATIC APPROXIMATIO An teratve method usng a separable quadratc approxmaton of the loss functon s easly obtaned by fxng at each teraton one value P n each quadratc term nvolvng two such values P and P j. Such an approxmaton s much more precse than the one obtaned by fxng both values n all such terms dscussed above. Then the algorthm of the prevous secton s as follows: (a) Obtan an ntal soluton P ( 0) P ( 0)... P ( 0) deletng all quadratc terms P B P wth j j j by solvng the problem after j n the loss functon (or whch s better by some heurstcs see below). Set the teraton counter k =. (b) Compute the followng separable quadratc approxmaton of the loss functon ( ) + k BP PB jpj + B0 P + B 00 (9) = = j= = j (c) Solve the problem wth the approxmate loss functon (9) to obtan a soluton ( k) ( k) ( k)... P P P (for ths purpose use DP explaned n the prevous secton). (d) Compute the approxmate and exact losses usng ( k) ( k) ( k)... P P P n (9) and (4). If these values dffer n absolute value by less than ε stop. Otherwse ncrease k by and return to (b). A better ntal approxmaton to the loss functon may be obtaned by computng a feasble ntal soluton by heurstcs such as followng: (h) Set P = P for =... ; mn PR P (h) Compute δ = = P ; (h3) If P + δ P > P for some set max reduce by. Return to (h); (h4) Set P δ P for all remanng. + P = P delete that ndex value and max The am of ths heurstcs s to obtan quckly a feasble soluton n whch the productons above the mnmum ones of the varous unts are as close as possble. Therefore t frst attempts to gve an equal load to each unt. If some upper bound s not satsfed the largest possble load s assgned to the correspondng unt. The procedure s terated wth the total unassgned load. More sophstcated heurstcs could take cost functons nto account but ths does not appear to be necessary. Example. (Wood and Wollenberg [6] p 7-0)). The fuel cost curves for the three unts n the sx-bus network consdered are gven by
64 P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 3 3 3 3 F ( P ) = 3. +. 669P + 0. 00533P F ( P ) = 00. 0 + 0. 333P + 0. 00889P F ( P ) = 40. 0 + 0. 833P + 0. 0074P wth a total load to be suppled P R = 0 and unt dspatch lmts 50. 0 P 00 37. 5 P 50 45. 0 P3 80. The coeffcents B00 = 4. 04 B0 = ( 0. 07660 0. 0034 0. 0890) ( = 3) and the B matrx s 0. 0006760 0. 0000953 0. 0000507 B =... 0 0000953 0 00050 0 000090. 0. 0000507 0. 000090 0. 000940 Soluton. The optmal soluton s obtaned n 3 teratons: It. F P P P 3 PR + P L Error 346.8 70.30 76.03 7.80 7.583 0.48846 365.09 7.5868 73.9075 73.569 9.063 0.0805 3 364.86 7.565 73.9348 73.5485 9.046 0.000 Example 3. (Lang and Glover [4]). The operatng costs of the generators are represented by the followng polynomals of thrd order: 3 a0 + ap + ap + a3p = 3 where coeffcents as well as P mn and P max are lsted below: Unt a 0 a a a 3 P mn P max.00 5.038 0.006490 0.00000333333 00 500 63.000 3.0000 0.0305740 0.00003333333 00 500 3 47.44 4.8997 0.00030845 0.0000007677 00 000 The load s P R = 400 and the losses are gven by The B -matrx s PL = 0. 000075P + 0. 00005P + 0. 000045P 3. 0. 0000750 0. 000005 0. 0000075 B =... 0 0000050 0 00005 0 000000. 0. 0000075 0. 00000 0. 0000450
P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm 65 Soluton. If an ntal soluton s obtaned by neglectng the off-dagonal elements of matrx B the optmal soluton s obtaned n four teratons: It. F P P P 3 PR + P L Error 659.0 359.704 406.478 665.957 43.4 0.46985 664.3 360.66 406.97 676.45 443.34 0.075683 3 664.68 36.36 406.969 675.079 443.4 0.00587 4 664.69 360.93 406.967 676.58 443.4 0.000366 Another way to derve an ntal soluton s also developed above n Secton 5. In ths example the ntal soluton obtaned s P = P = 433. 333 ; P 3 = 533. 333. The optmal soluton s then reached n only two teratons: It. F P P P 3 PR + P L Error 664.60 363.9 407.547 67.448 443. 0.9736 664.68 359. 406.959 677.44 443.4 0.000977 Example 4. In the prevous examples convex programmng problems are solved snce both the objectve functon F and the transmsson losses P are convex. ote that T convexty of the P L functon follows from the postve defnteness of the B matrx (.e. the man mnors of B are postve n Examples and 3 above). In ths example we change elements b and b from Example 3.e. we now have b = b = 0. 00005. Then the mnor of the second order s negatve (det = 0. 000075 0. 00005 0. 00005 < 0) and thus there are more than one local mnma. Snce the penalty factor method s derved from the frst order condtons t s a local optmzaton method and may not reach the global optmum. Soluton. Wth our separable approxmaton dynamc programmng method the followng locally optmal soluton s obtaned n sx teratons: It. F P P P 3 PR + P L Error 6699.85 375.87 00.875 999.538 475.60 8.50537 6657.0 49.456 98.993 946.974 465.4 0.5955 3 6684.7 348.96 40.648 703.57 453.0 3.97778 4 6703.30 38.46 403.99 75.495 457. 0.34585 5 670.74 37.45 404.04 75.39 456.83 0.009888 6 670.69 37.437 403.99 75.39 456.8 0.000 We then ran the penalty factor method to see what soluton t gves for ths non-convex example. For that purpose we used software attached to the book [6]. The soluton obtaned s as follows: FP ( P P 3) = 6940. 35 ; P = 48. 6 P = 00. 0 P 3 = 000. 0 λ = 4. 6765. Therefore the objectve functon value and the total losses are 6940.35 and 8.6 respectvely (compare those values wth 670.69 and 56.8 obtaned by our approach). L
66 P. Hansen. Mladenov} / A Separable Approxmaton Dynamc Programmng Algorthm We also tred a modfcaton of the penalty factor method: nstead of adjustng λ n the nner loop (as suggested n [6] for Step 3 n the algorthm gven n Secton 3) we smply fnd the closest soluton that satsfes the range constrants (3) as well. In other words wthn Step 3 we teratvely project the current soluton ( P... P n) onto the hyperplane defned by () and the hypercube (3) untl a feasble soluton s reached. ote that both methods are equvalent f the pont obtaned after solvng the coordnaton equatons s feasble. The result obtaned n 5 teratons s: F( 9. 43. 79 000. 00) = 68. 3. Agan ths soluton s worse than the dynamc programmng one. 6. COCLUSIO A new method for statc termal power unts economc dspatch problem wth transmsson losses s proposed. It uses a separable quadratc approxmaton to the loss functon and refnes t teratvely. Convergence s qucker than wth prevous methods. Moreover t may lend to better locally optmal solutons than those methods n case of non-convexty of the loss functon. REFERECES [] Bellman R.E. and Dreyfus S.E. Appled Dynamc Programmng Prnceton Unversty Press Prnceton 96. [] Chowdury B.H. and Rahman S. "A revew of recent advances n economc dspatch'' IEEE Trans. PWRS 5 (V) (990) 48-57. [3] Krchmayer L.L. and Stagg G.W. "Evaluaton of methods of coordnatng nternal fuel and ncremental transmsson losses'' AIEE Trans. 7 (III) (95) 53-50 [4] Lang Z.-X. and Glover J.D. "A zoom feature for a dynamc programmng soluton to economc dspatch'' IEEE Trans. PWRS 7 (II) 99 544-549. [5] Rnglee R.J. and Wllams D.D. "Economc dspatch operaton consderng valve throttng losses II - Dstrbuton of system loads by method of dynamc programmng'' IEEE Trans. PAS 8 (I) (963) 65-6. [6] Wood A.J. and Wollenberg B.F. Power Generaton Operaton and Control Wley ew York 984. nd revsed edton 996.