Scenario Reduction Algorithm and Creation of Multi-Stage Scenario Trees

Transcription

1 Fakulteta za Elektrotehiko Heike Brad, Eva Thori, Christoph Weber Sceario Reductio Algorithm ad Creatio of Multi-Stage Sceario Trees OSCOGEN Discussio Paper No. 7 Cotract No. ENK5-CT Project co-fuded by the Europea Commuity uder the 5 th Framework Programme ( Cotract No. BBW Project co-fuded by the Swiss Federal Agecy for Educatio ad Sciece February 2002 Cotract No. CEU-BS-/200 Project co-fuded by Termoelektrara toplara Ljubljaa, d.o.o.

2 SCENARIO REDUCTION AND CREATION OF MULTI-STAGE SCENARIO TREES Sceario reductio Creatio of a Multi-Stage Sceario Tree EXAMPLE Scearios ad parameter values Deletio of complete scearios Trasformatio ito a Multi-Stage Tree LITERATURE... 4

3 Sceario Reductio ad Creatio of Multi-Stage Sceario Trees Whe usig Mote Carlo simulatios, a huge umber of differet scearios is created accordig to the assumed probability distributios of the stochastic parameters of a give multi-stage optimisatio model. For such high umbers of scearios it is impossible to umerically obtai a solutio for the multi-stage optimisatio problem of the uit commitmet. Moreover, the sceario tree cosistig of these scearios is oly a oe-stage tree. Thus, strategies for reducig the umber of scearios have to be studied to eable fidig a umerical solutio of the problem as well as algorithms for costructig a multistage sceario tree out of a give set of scearios. Simply geeratig a very small umber of scearios by Mote Carlo simulatios is ot wated sice less scearios give less iformatio. Ideed, the aim is to loose oly a miimum of iformatio by the reductio process applied to the whole set of scearios. Actually, two steps will have to be doe: first, the pure umber of scearios will have to be reduced. Afterwards, based o all remaiig scearios still formig a oe-stage tree, a multi-stage sceario tree is costructed by deletig oly ier odes ad creatig brachig withi the sceario tree.. Sceario reductio I the mathematical literature some algorithms are proposed for reducig a give set of scearios ad costructig a sceario tree based o the idea that the reduced sceario tree i a give sese still is a sufficiet good approximatio of the origial oe /Dupacova et. al. 2000/. Therefore, the so called Katorovich distace D KA ( P, Q of a probability distributio P of a give umber of scearios ad a distributio of scearios Q with give probabilities for each sceario /Rachev 99/ has to be cosidered. I the special case, that for Q a subset of all scearios is chose together with their probabilities, i.e. Q is a reduced probability distributio for P, a optimal probability distributio Q based o these scearios ca be costructed possessig a miimal Katorovich distace to P /Dupacova et. al. 2000/. The problem of determiig the best subset of the origial set of scearios ca be solved by formulatig a travellig salesma problem ad ca thus ot be solved i polyomial time. A heuristic is used for fidig the scearios to be deleted from all scearios /Dupacova et. al. 2000/. I the followig the reductio algorithm is described i detail. Let deote the umber of stages of the optimisatio problem ad the umber of scearios. It is assumed that all scearios have a commo root i a oe-stage tree where brachig occurs oly after } (i the root ode. A sceario, i {, Κ, S, is defied as a sequece of odes of the tree S T

4 where η ( η, η Κ,, i =, Κ, S = η 0 0 = η0 =, Κ T i, S (i deotes the root of all scearios ad η T deotes the leaf of this sceario i withi the tree. ( j ( j A ode belogs to sceario j, Κ, o stage s, Κ,. For each ode, has { s 0, Κ p s ( j, a vector p R s of parameters is give. Each ode o stage s 0, Κ, parameters. The probability to get from stage j to stage j+ withi sceario i, i.e. to (i (i get from to, is deoted by. Thus the probability for the whole sceario is give by η s, T η j } η j+ s p { S j, j + = T j= 0 } } j, j+ = (i ( j The distace betwee two scearios ad is defied as d ( ( T i j ( j ( ( p p / 2 2, = s s s= 0 0, { T } η s { } accordig to a orm i the space of the parameter vectors. So i a first step, the algorithm for deletig whole scearios is described i the followig. This deletig procedure is applied iteratively, deletig oe sceario i each step ad cosequetly chagig the probabilities of other scearios, util a give umber of scearios is remaiig. Determie the sceario to be deleted: Remove sceario ( s s {, Κ,S ( m ( ( m mi d(, = mi { mi d(, } m {, Κ, } m S T satisfyig Ituitively it is clear that oe tries to delete scearios that are, accordig to the defied distace, ear to some other sceario; otherwise, possibly importat iformatio might be lost deletig a sceario that sigificatly differs from all the others. Not oly cosiderig distaces, but also the probabilities of the scearios, those scearios havig a small probability are more likely to be deleted tha others. 2 Chage the umber of scearios: S : 3 Chage the probability of the sceario = S (s, that is the earest to ( s : Set d (, = mi d(, : = + 0, 0, 0,

5 This has to be doe, as the sum of all probabilities of the remaiig scearios should remai equal to ad the oly brachig occurs at stage 0 at the root ode. 4 Cotiue with step as log as S > N. Otherwise STOP..2 Creatio of a Multi-Stage Sceario Tree Havig doe this full iterative deletio of sigle scearios util the desired umber of scearios remais, oe is still left with a sceario tree cosistig of oly oe stage. Thus i the followig, a algorithm is described used for creatig a multi-stage sceario tree by deletig ier odes without chagig the umber of leaves of the tree. The procedure described i. forms the basis for this algorithm that additioally has to deal with odes of the scearios ad successor sets for these odes. This algorithm is a variatio of a algorithm preseted i /Gröwe-Kuska 200/. This algorithm proceeds iteratively i two ways: firstly, by applyig it o a fixed stage of the sceario tree to be costructed for a a priori kow umber of times, ad secodly, by recursively goig backwards the stages of the sceario tree util the secod stage of the tree is reached. Kowig the startig value of the umber of scearios ad the umber of stages the fial sceario tree is to possess, it is a priori fixed how ofte this algorithm is to be applied o each stage. We ow itroduce the otatio of a series of odes up to a give stage replacig the otatio of a sceario i cases i which this algorithm is applied o earlier tha the last stage of the sceario tree. Therefor some additioal otatio has to be itroduced: A series of odes up to stage s = > 0 is ( s ( η0, η, Κ, η The distace betwee two series of odes both cotaiig + odes is defied to be I case of d ( j ( j ( ( p p 2, = s s s= 0. / 2. = T, the series of odes is still called a sceario T ( s ( η0, η, Κ, η = =. The set of odes o stage > 0 is deoted by S, iitially give by { : i {, Κ } S = η, The followig defiitio simplifies the hadlig of odes o a give stage of the tree to be costructed. Defiitio: A series of odes ( η0, η, Κ, η S m T S is called admissible if all odes except the ( s first oe belog to the actual sets of odes o all stages, i.e. η S for m m

6 {, Κ } m, (k (k We defie the set of successors of each ier ode j, SUCC η j, j {, Κ, }.This algorithm is applicable for trees with a fixed umber of successors of T η ( each ier ode (e.g. biary sceario trees or with a free structure with o such restrictios. I the importat but special case of a biary sceario tree, oe would demad ( k ( j 2 j {, Κ, T } k {, Κ S SUCC η, }. This ca become importat whe fidig ew predecessors for odes. Iitially, startig with the scearios without brachig at ier odes of the tree, each ier ode has precisely oe successor (the leaves have o successor while the root ode has the algorithm. S successors at the start of The algorithm proceeds as follows: Set := T to be the actual stage. 2 Determie the series of admissible odes = ( η0, η, Κ, η {, Κ } s, S ( whose ier odes η k = 0 s, Κ, η are to be deleted ad afterwards ewly set: ( m ( m ( l (, = mi k, k + mi d ( { } k, k + mi d, m, Κ, S m l k = 0 This is doe i the same way as i the algorithm described above: we try to elimiate the ier odes of those scearios that are the most similar to aother oe. But here, sice o ode at the actual stage is deleted, oly the distaces of the first odes of each sceario are take ito accout. Therefore the iformatio about the ode at stage is ot importat for deletig the ier odes that are o the stages up to. This is differet to what is proposed i /Gröwe-Kuska 200/, where the whole distace up to stage is used as a criterio for deletio of ier odes. ( s 3 Delete the ier odes η, Κ, η from the set of odes o each stage of the tree: S { } ( s η m : = for m {, Κ, }. This is the importat differece to the m S m algorithm of deletio of full scearios; here, the ode at the actual stage of the sceario remais admissible ad oly all those odes lyig o the way from the root to this ode are deleted. ( 4 Determie the ew predecessors of the last ode s ( : Sice the odes η are o loger admissible, we have to chage these odes of sceario determie the series of admissible odes up to stage, = η0, η η ( s s, Κ, η. First we (s (, Κ, η, that

7 has the smallest distace to = ( η0, η,, η Κ. I case of restrictios about the umber of successors of ier odes, these have to be cosidered i fidig the miimum o the right had side of d ( m ( m (, = mi { d (, : s( η < 2} m s ( The odes of sceario s o the stages, Κ, are the chaged to equal the correspodig odes η Κ,. This meas that the two series of odes ad, η s are merged (they become idetical, ad that at stage a brachig occurs ito the two successors (s η ad : the ew series of predecessors of is thus (s (s η η 0, η,, ( η Κ η. So a ew way from root to s is defied based o admissible odes. s 5 Chagig the probabilities to reach ad accordig to ad, ( η 0, η ( 0, 0, : = / +, 0, (s η from their commo predecessor ( 0, 0, : = / + sice the probability to reach η from the root ode has to be chaged to 0, 0, : = + 0, η because of mergig the ier odes ( s η l ad η for l =, Κ,. (s l The reaso is that the probabilities for reachig ( s η (s ad η from root must ot chage i order to coserve the sum of the probabilities for reachig all admissible odes o the actual stage from the root ode. 6 Retur to step 2 as log as more ier odes have to be deleted at the actual stage 7 Set : = ad go to step 2. 2 Example 2. Scearios ad parameter values This complex algorithm will be made clear by studyig a example i detail. The aim is to costruct a biary three-stage sceario tree based o 0 scearios give with a commo root ode ad i total 6 parameter values for each of them. The values of the parameters of all scearios are listed i Table 2-.

8 Table 2-: Give scearios T0 T T2 T3 T4 T5 S S S3 8 S S S S S S S The secod ad third time step (ad thus also their parameters are both cotaied withi stage oe ad the fifth ad sixth time step are cotaied withi stage three. The first time step has the same parameter value for all scearios sice it is the commo root ode to all scearios. The tree i its iitial form is show i Figure 2-. 0, -0 36, ,36 S 0, -04 5, ,5 S2-07, , S3 0, 0, -0 6, ,6 S S5 8 0, 0, 0, S6 S7 0, S8-25 0, ,0 S9 0, S0

9 Figure 2-: Origial sceario tree with 0 scearios 2.2 Deletio of complete scearios First the umber of scearios to be deleted has to be calculated. For obtaiig a biary tree, two scearios have to be completely deleted i order to have eight remaiig scearios for a three stage tree. Therefore i a first step the distace matrix show i Table 2-2 is calculated. A additioal colum is added i Table 2-2 cotaiig the value of the probability of the sceario times the miimal distace to the other scearios: mi d (, Table 2-2: Distace matrix for the origial tree mi d(, S S2 S3 S4 S5 S6 S7 S8 S9 S0 S S S S S S ? S S S S From the etries of the distace matrix, oe fids i a first step that sceario 3 is deleted as the expressio mi d(, takes the smallest value for s = 3. As sceario 3 has the smallest distace to sceario 6 it will be merged with S6 ad the probability of the sixth sceario is chaged by addig the probability of S3. The distace matrix after deletio of sceario 3 is show i Table 2-3. (It could as well be the other way aroud, that S6 is deleted ad merged to S3, or? Table 2-3: Distace matrix after deletio of sceario 3 S S2 S4 S5 S6 S7 S8 S9 S0 mi d(, S S S S S S

10 S S S I a secod step, sceario S4 is deleted sice the distace to S6 is the smallest of all distaces ad S4 is less probable tha S6 (remember that the probability of S6 was icreased due to the mergig with S3 i the first step. The emergig sceario tree with eight remaiig scearios is show i Figure , 36, ,36 S -04 5, ,5 S2 0, -00 0, S5 8 0,3 0, 0, S6 S7 0, S8-25 0, ,0 S9 0, S0 Figure 2-2: Tree after deletio of scearios S3 ad S4 2.3 Trasformatio ito a Multi-Stage Tree Now we are ready to trasform this oe-stage tree with eight scearios ito a biary tree followig the steps of the algorithm described above. The etries of the distace matrix o the secod stage, listed i Table 2-4, of this tree show that the scearios 2 ad 9 have the smallest value of probability multiplied by the miimal distace to the other series of odes up to stage 2 (see last colum i Table 2-4. Both S2 ad S9 havig equal probability, oe may choose to

11 delete the ier odes of sceario 2. As stated above, sice the odes at stage 3 remai, it is ot ecessary to cosider distaces of series of odes up to stage 3. Table 2-4: Distace matrix o stage 2 k= 0 S S2 S5 S6 S7 S8 S9 S0 k, k+ mid2 ( 2, 2 S S S S S S S S The sceario S2 is give by the series of odes (2 = ad the sceario S9 by ( 00, 04, 05, 06 ( 00, 25, 26, 27 (9 =. We ow have to fid the ew predecessor of ode -06 by lookig at the distaces betwee ode -05 (the origial predecessor of -06 ad the other odes o stage two, that all are admissible. These values are give i the left part of Table 2-4 i the row of S2, idicatig that ode -26 of sceario 9 will become the ew predecessor of ode -06, sice its distace to -05 of S2 is the smallest of all admissible odes of stage 2. Thus the ew series of ode of sceario S2 looks like: (2 = ( 00, 25, 26, 06 Chagig the probabilities is doe afterwards as described i the algorithm above with the old (2 (9 values of ad : 0, 0, (2 (2 2,3 = 0, / (2 (9 ( : + (9 (9 2,3 = 0, / 0, 0, (2 (9 ( : + ad afterwards settig the ew values for reachig ode -25 from ode -00: 0, 0, : = (9 (2 (9 (2 0, = 0, + 0,, 0, : Cosequetly, ode -26 has two successors ad is o loger cosidered i the search for ew predecessors i the followig. The resultig tree is show i Figure 2-3. (9 0,

12 , ,36 S 0, -06 5,5 S2-00 0, S5 8 0,3 0, S6 0, S7 0, S8-25 0, ,0 S9 0, S0 Figure 2-3: Tree after sceario 2 has merged with sceario 9 I the same way, the ier odes of three further scearios will be deleted. Note, that of course the values of sceario 2 i the distace matrix have ot to be cosidered aymore as well as the values of the other scearios whose ier odes are already deleted. The ier odes of sceario 7 are deleted secodly, fidig ode -23 (S8 as ew predecessor of ode -2 (S7. After that the odes -0 ad -02 of sceario are deleted ad ode -4 (S5 becomes the ew predecessor of ode -03. Fially, the ier odes of S0 are deleted ad ode -7 (S6 is the ew predecessor of ode -30 (S0. I each case, the probabilities are adjusted i such a way preservig the origial values for reachig the leaf odes from root. Now the tree structure is show i Figure 2-4. Now we go to stage two ad determie two odes o stage that will be deleted (they are the ier odes of a series of odes from root to a admissible ode o stage two. The values show i the last colum of Table 2-5 idicate that ode -25 of sceario S9 is deleted o stage ad that ode -6 (S6 becomes the ew predecessor of ode -26 (left part of Table 2-5.

13 , ,36 S5 S -00 0, S8 8 0,4-2 S7 0, ,75-8 S6 0,25-30 S0-25 0, ,0 S9-06 5,5 S2 Figure 2-4: Structure of the sceario tree after creatig the secod stage with four remaiig odes The, i the secod step, oly scearios S5 (with probability 0.2, S6 (ow with probability 0.6 ad S8 (with probability 0.2 are available, ad oly the sub matrix of the first three rows ad colums of the distace matrix o stage (Table 2-5 eed to be cosidered. Clearly, the miimal product of probability ad miimal distace is the the probability of S5 times distace of S5 to S6 o stage, (the other values are ad for S6 ad S8, respectively. Thus, ode -3 is deleted ad the ew predecessor of ode -4 is to be foud. Table 2-5: Distace matrix o stage ( S5 S6 S8 S9 0, mid, S S S S Now, oly ode -22 (S8 is available sice -6 (S6 already has two successors ad -3 as well as -25 were already deleted o this stage. Fially, the probabilities are adjusted. So the fial structure of the biary three-stage sceario tree is show i Figure 2-5.

14 S ,36 S ,4 0, , , S8 S7 S6 0,333 0,25-30 S ,0 S9-06 5,5 S2 Figure 2-5: Fial form of the biary sceario tree

15 Literature /Dupacova et al. 2000/ Dupacova, J.; Gröwe-Kuska, N.; Römisch, W.: Sceario reductio i stochastic programmig: A approach usig probability metrics, Preprit 00-9, Istitut für Mathematik, Humboldt-Uiversität Berli, 2000, accepted for publicatio i Mathematical Programmig /Rachev 99/ Rachev, S. T.: Probability metrics ad the stability of stochastic models, J. Wiley ad Sos, New York, 99 /Gröwe-Kuska et al. 999/ Gröwe-Kuska, N.; Nowak, M. P.; Römisch, W.; Weger, I.: Optimierug eies hydrothermische Kraftwerkssystems uter Ugewissheit, i: Optimierug i der Eergieversorgug. Plaugsaufgabe i liberalisierte Eergiemärkte, VDI- Berichte 508, Düsseldorf 999, /Gröwe-Kuska 200/ Gröwe-Kuska, N.: Modellierug stochastischer Dateprozesse für Optimierugsmodelle der Eergiewirtschaft, Talk at VDI-Tagug Optimierug i der Eergiewirtschaft, Oktober 200, Veitshöchheim