Home

Stochastic Programming and Tradeoff Analysis in TIMES

1. Input parameter Related Units Ranges amp l Instances l Description Affected equations or 1 2 Default values amp Default Required Omit Special variables Indexes parameters A 1 E inter extrapolation conditions SW START SW SUBS e Year If less than or equal to start Start year of stochastic e All 0 SW_SPROB e BOT EOT integer of previous stage stage is stage j SW_PROB default value see doc combined with previous SW_LAMBDA e Default i e n a SW _SUBS See above Dimensionless Can be left unspecified if Number of sub states of e All jw 1 64 default value specified for first state the world for state w at see Instances which is then the default stage j SW SPROB See above e Dimensionless Can be left unspecified see Conditional probability e All w e 0 1 doc for details of sub state w at stage j default value none given its parent s state SW _PROB See above e Dimensionless Use instead of SW SPROB Total probability of e All w e 0 1 mainly useful for MARKAL stochastic scenario default value none style 2 stage programs only SW_LAMBDA See above e Dimensionless Optional use only if risk Risk aversion e EQ OBJ 0 INF aversion is to be included coefficient The first row contains the parameter name the second row contains in brackets the index domain over which the parameter is defined This column gives references to related input parameters or set
2. Parameter Description SPAR_PEAKM w r t cg s Marginals for the peak equations SPAR_UCSM w uc Marginals for the user constraint equations slacks Capacity bound parameters SPAR_CAPUP w r t p Upper bound on overall capacity in a period SPAR_CAPLO w r t p Lower bound on overall capacity in a period Climate module result parameters CM SRESULT w item t Basic results from stochastic Climate Module CM SMAXC M w y Shadow price of climate variable constraint The reporting parameters for the Climate Module correspond to the same ones in the standard mode with the adjunction of the prefix CM S 4 4 Parameters for tradeoff analysis The two phase tradeoff analysis facility is available under the stochastic mode only The stochastic mode should therefore be activated when using the tradeoff analysis facility If no SOWs are explicitly defined by the user the model will be run only once SOW 1 using the two solution phases described above in Section 3 However usually the user would like to estimate a full tradeoff curve consisting of a number of discrete solution points S OWz1 N The number of points in the curve i e the number of S OWs should be defined by SW SUBS 1 1 N The parameter attributes that can be varied in such sensitivity analyses are the same uncertain parameters which can also be used for multi stage stochastic programming The following uncertain attributes are perh
3. Units Ranges amp Instances Description Affected equations or pue parameter Related Default values amp Default Required Omit Special variables Indexes parameters i 3 pa inter extrapolation conditions S CAP BND cap bnd e Capacity unit Since inter extrapolation is Bound on total capacity e Causes the generation r datayear p bd j w ncap bnd e 0 INF by default only migrated the of a process in a period of a capacity transfer spar_caplo default value none bound must be specified for equation EQ CPT spar capup e Default i e migrate to each period desired if no Imposes a bound on the period year specific option regarding capacity variable inter extrapolation is given VAR_CAP S_DAM_COST dam_cost e Currency units per None Damage cost on EQ_OBJ r y com cur j w commodity units commodity net VAR_COMNET e 0 INF production e Default i e standard S_CM_MAXC cm_maxc e GtC C Since no inter extrapolation Maximum allowed Causes the generation y item j w e 0 INF is done by default the bound atmospheric CO of a concentration default value none must be specified for each concentration in GtC bound equation e Default i e none period desired if no specific in a given year EQ CLIMAX option regarding inter extrapolation is given S CM CONST cm const e None Item must be either Climate module cons e Temperature equations item j w SIGMA or CS tants
4. analysis A N E N E N Ti 96903 OOOO Figure 2 Possible set ups for sensitivity tradeoff analysis 2 MULTI STAGE STOCHASTI C PROGRAMMI NG 2 1 General The key observation is that prior to resolution time the decision maker and hence the model does not know the eventual values of the random parameters but still has to take decisions On the contrary after resolution the decision maker knows with certainty the outcome of some event s and his decisions will be different depending of which outcome has occurred For the example of figure 1 in 2000 and 2005 there can be only one set of decisions whereas in 2010 there will be two sets of decisions contingent on which of the Mitigation outcomes High or Low has occurred and in 2015 2020 2035 there will be four sets of contingent decisions This remark leads directly to the following general multi period multi stage stochastic program 1 3 below The formulation described here is based on Dantzig 1955 Wets 1989 or Kanudia and Loulou 1998 and uses the expected cost criterion Note that this is a LP but its size is larger than that of the deterministic TIMES model Minimize Z Y Cts x X t s x pits 1 teT seS t Subject to A t s x X t s 2 b t s 2 DG g t s xX tg t s 2e s Vse S T 3 teT where t time period T set of time periods s state index SH set of state indices for time period t
5. 3 Use the parameter SW SPROB to specify the conditional probabilities of the sub states of each SOW at each stage j Alternatively the total probabilities of each SOW at the last stage can be specified by using SW PROB The use of SW SPROB is the recommended method for which the following rules apply e If the parent SOW is subject to cloning SW SPROB can be left unspecified and it will then inherit the probabilities from the sub states of the first SOW at the previous stage e If SW SPROB is not specified and the parent S OW is not subject to cloning SW SPROB will be automatically assigned a probability UP m where UP is the unassigned probability among the sub states of the parent S OW and m is the number of remaining sub states for which probabilities are to be assigned An even probability distribution among the sub states can thus be specified without using any SW SPROB for non cloned SOW or by specifying the probability for the first sub state only for cloned S OW 4 Use the parameter SW LAMBDA to specify the risk aversion coefficient If not specified the objective function represents the expected total discounted system costs without risk aversion A quick reference of the use of the control parameters is given in Appendix A 5 3 2 Specification of uncertain parameters The uncertain parameters shown in Table 11 can be currently used in stochastic TIMES A quick reference of the use of the input parameters is given in
6. Appendix B Apart from the special aspects concerning the relative type the use of the uncertain parameters is basically similar to the corresponding deterministic parameters However the index of both the stage and state of the world has to be specified when using these parameters The same default interpolation rules are applied to the uncertain parameters as to the deterministic counterparts Relative parameters are applied as multipliers to the corres ponding deterministic baseline parameters and cumulatively over stages Any relative parameters defined at an earlier stage of the event tree are included in the combined multipliers at a later stage The user can also utilize the cloning facility for the automatic copying of uncertain parameters from the sub states of the first state at each stage to the sub states of other states As mentioned above these other states are subject to cloning only if SW SUBS was left unspecified for them 5 4 Sensitivity analyses The stochastic mode can also be used for a series of deterministic runs This can be accomplished by combining all or some of the stochastic stages with the first stage An easy way to do this is to specify for the first stage a value of SW START larger than all or some of the subsequent SW START Consequently the subsequent stages will have a value of SW START less than or equal to the first stage and are therefore combined with the first stage In effect this will mean that
7. DEM 4 1 1 05 MEG 2015 DEM 4 2 0 95 REG 2015 DEM 5 1 1 04 REG 2015 DEM 5 2 0 893 7 After running the stochastic model the user might wish to compare the results with equivalent deterministic scenarios This can be accomplished by changing the start years so that the first stage has a later start year than all other stages see section 4 4 PARAMETER SW START 1 9999 2 2010 3 2020 5 6 Tradeoff analyses 5 6 1 Activating the tradeoff analysis mode The tradeoff analysis facility has to be activated by either one of the following settings in the run file SET STAGES YES SET SENSIS YES The first alternative simply activates the stochastic mode which is required when using the tradeoff facility The second alternative additionally enables the use of the active basis information successively between solving each point of the sensitivity analysis utilizing the so called warm start method In some cases this may significantly improve the solution speed because the solution of each point can start from the optimal solution of the previous point However in cases where the successive model instances to be solved differ considerably from each other this may not be an efficient option The judg ment of whether to use the warm start or not has to be made by the user The set ups D and E described in the introduction are only possible with the setting STAGES YES When using the stochastic mode for tradeoff analysis there
8. OBJ 1 by n i and go back to the economic equilibrium by re optimizing with the original objective function i e the total discounted costs or surplus phase 2 One can use the multiphase tradeoff analysis for complex multi objective analyses 5 7 Tradeoff analysis examples Example la Single two phase tradeoff analysis Possible use 1 above without iteration over different deviation bounds Assume that the generic row to be optimized in Phase 1 is UCEXT which should be minimized Activate stochastic mode SET STAGES YES Define objective function to be minimized in Phase 1 S UCOBJ UCEXT 1 1 Define a 10 deviation bound to be applied in Phase 2 UC RHS UCEXT N 0 1 Example 1b Two phase tradeoff curve analysis Possible use 1 above with iteration over 5 different deviation bounds Assume that the generic row to be optimized in Phase 1 is UCEXT which should be maximized Activate stochastic mode and define number of tradeoff points SET STAGES YES SW_SUBS 1 1 5 Define objective function to be used in Phase 1 applied to all 5 points S UCOBJ UCEXT 1 2 1 Define the deviation bounds to be applied in Phase 2 10 50 S UC RHS UCEXT N 271 0 1 S UC RHS UCEXT N 2 2 0 2 S UC RHS UCEXT N 27 3 0 3 S UC RHS UCEXT N 274 0 4 S UC RHS UCEXT N 275 0 5 Example 1c Two phase tradeoff curve analysis Possible use 1 above with iteration over 5 different slo
9. SIGMAI or CS S_FLO_CUM flo_cum e Flow activity unit None Bound on cumulative EQ CUMFLO r p c yl y2 bd j w act cum e Default value none flow or activity S FLO FUNC flo func e None None Multiplier for process e EQ PTRANS r datayear p cgl cg2 flo sum act eff e Default value 1 transformation coeff EQ_ACTEFF jw e Default i e standard S NCAP COST ncap cost e Currency units None Multiplier for process e EQ OBJINV r datayear p j w Default value 1 e Default i e standard investment cost Units Ranges amp Instances Description Affected equations or Inp rparamcten Related _ Default values amp Default Required Omit Special variables deses parameters inter extrapolation conditions S UC RHS uc rhs None Used in user constraints RHS constant with e User constraints uc_n lim j w open default value bound type of lim of EQ D UC none a user constraint Default i e none S UC RHSR uc rhsr None Used in user constraints RHS constant with User constraints r uc_n lim j w open default value bound type of lim of EQ I UCR none a user constraint Default i e none S UC RHST uc rhst None Used in user constraints RHS constant with User constraints uc n y lim j w open default value bound type of lim of EQ _UCT none a user constraint Default i e migrate to period year S_UC_RHSRT uc_rhsrt None Used in user constraints RHS constant with e U
10. by VAS So for example the capacity variable VAR_ CAP r t p becomes VAS _ CAP r t p sow During matrix generation the appropriate SOW index value is then entered into VAS CAP according to the set SW T and the period being worked on Table 7 Variables for stochastic TIMES Variable Description The variable equal to the sum of the total discounted system cost associated with each SOW The variable equal to the total objective function in Phase 1 of the Tradeoff Analysis facility not used under multi stage stochastics The variable equal to the expected value of the total discounted System cost The upside deviation between the total system cost for each SOW and the expected value of the total system cost VAS UC OBJZ w VAS UC OBJ1 w VAS EXPOBJ VAS UPDEV w As there is thus essentially no redefinition of the variables for the stochastic formulation other than the control of the instances of the variable according to the control sets SW T and SW TSW the user is referred to Chapter 4 of the TIMES Reference Manual for details on the variables of the model Below in Table 7 the variables strictly involved with the stochastic version are listed however as it is rather straightforward the description of the variable details is not repeated here for the stochastic variables 4 6 Equations As noted earlier and as is the case with the variables the equations that are used to model stochastics a
11. example is said to have three stages i e two resolution times The simplest non trivial event tree has only two stages a single resolution time 6 um e n High Growth 0 4 High Mitigation a x 1O x x Low Growth gt N High Growth Low Mitigation 0 5 OD gt gt gt Low Growth i ere or on e Stage 1 Stage 2 1995 2000 2005 2010 2015 2020 2025 2030 2035 Figure 1 Event Tree for a three stage stochastic TIMES Example While stochastic programming is an advanced way to take into account uncertainties a more common and very useful way to analyze the impact of uncertainties is sensitivity analysis In sensitivity analysis the values of some important exogenous assumptions are varied and a series of model runs is performed over a discrete set of combinations of these assumptions Sensitivity analysis is often combined with tradeoff analysis where the tradeoff relation between several objectives is analyzed The stochastic mode provides an efficient tool for both sensitivity and tradeoff analyses because it enables the use of so called uncertain attri butes The uncertain attributes are similar to the corresponding standard TIMES attributes but they can be defined over a discrete set of states of the world SOW In stochastic prog ramming the SOWs correspond to the branches of the event tree but they can equally well be used for sensitivity analysis so that the
12. in Figure 1 can be specified as follows PARAMETER SW START 2 2010 PARAMETER SW SUBS I adl oh az c2 PARAMETER SW SPROB 2 10 5 3 10 4 3 3 0 515 Assume that High Mitigation in Figure 1 corresponds to a constant CO concentration limit of 770 between 2010 and 2035 and Low Mitigation corresponds to the limits of 790 and 950 in 2010 and 2035 respectively The mitigation parameters can then be specified as follows the year index 0 is a placeholder for the interpolation control option PARAMETER S CM MAXCO2C 0 2 1 1 2010 2 1 770 0 2 2 1 2010 2 2 790 Assume that High Growth is 5 higher than the baseline projection and Low Growth is 5 lower The High Low growth parameters can then be specified as follows assuming that the region is REG and the demand is DEM PARAMETER S COM PROJ REG 2015 DEM 3 1 1 05 REG 2015 DEM 3 2 0 95 REG 2015 DEM 3 3 1 05 REG 2015 DEM 3 4 0 95 Table 3 GAMS control variables for stochastic TIMES Control Value of control variable variable Standard Under stochastics EQ EQ ES VAR VAR VAS VART VAR SUM SW TSW SOW T W VAS VARM VAR SUM SW TSW SOW MODLYEAR W VAS VARV VAR SUM SW TSW SOW V W VAS SOW d SOW SWD mm WW SWTD ULWW SWT x SW T T SOW SWS EN W Table 4 Internal sets and parameters for stochastic TIMES Set Description SW CHILD j w w Child sub states of the world for each SOW at each stage SW COPY jw SOWS a
13. s at every period t whereas there was only one variable X 1995 1 in the previous formulation Minimize Z Y Y C t s x X t s x p t s teT seS t Subject to A t s X X t s b t s all t all s 2 oDe sxXq s 2e s alt alls 3 teT Of course in this approach we need to add equality constraints to express the fact that some scenarios are identical at some periods In the example of Figure 1 we would have X 1995 1 X 1995 2 X 1995 3 X 1995 4 X 2000 1 X 2000 2 X 2000 3 X 2000 4 X 2005 1 X 2005 2 X 2005 3 X 2005 4 X 2010 1 X 2010 2 X 2010 3 X 2010 4 Although this formulation is less parsimonious in terms of additional variables and constraints many of these extra variables and constraints are in fact eliminated by the pre processor of most optimizers The main advantage of this new formulation is the ease of producing outputs organized by scenario In the current implementation of stochastic TIMES the first approach has been used Equations 1 3 The results are however reported for all scenarios in the same way as in the second approach 2 2 Alternative objective formulations The preceding description of stochastic programming assumes that the policy maker accepts the expected cost as his optimizing criterion This is equivalent to saying that he is risk neutral In many situations the assumption of risk neutrality is only an approxi mation of the true utility function of a decision maker
14. stage j The uncertain parameter has not been specified for some of the sub states of S OW which are thus considered subject to cloning Table 2 Current set of uncertain input parameters for stochastic TIMES Parameter Description Type S COM PROUJ r y c j w Demand projection Rel S CAP BND r y p l j w Bound on total installed capacity Abs S COM CUMPRD try y c l j w Cumulative bound on commodity production Abs S COM CUMNET ry y c l j w Cumulative bound on commodity net prod Abs S_FLO_CUM r p c y y j w Cumulative bound on flow or activity Abs S FLO FUNC ry p cg cg2 j w Process transformation efficiency Rel S NCAP COST ry p j w Process investment cost Rel S UC RHS xxx lj w RHS constant of user constraint Abs S DAM COST ry c cur j w Damage cost of net production of commodity Abs S CM MAXC y item j w Bound on maximum level of climate variable Abs S CM CONST item j w Climate module constant CS or SIGMA1 Abs The last two indexes of all uncertain parameters are j stage and w state of world The stage index has been included in the parameters to ensure unambiguity without the stage index there could easily be ambiguities in the parameter values for earlier stages Note that demand projections are by default interpolated and extrapolated over all valid periods for each stage Bound parameters are by default interpolated within periods only Example The event tree of the example shown
15. OBJ UCEXT2 2 1 S_UCOBJ UCEXT3 3 1 S_UCOBJ UCEXT4 4 z 1 S_UCOBJ UCEXTS 5 1 Define a 10 deviation bound to be applied in Phase 2 in this example the same 10 deviation bound is applied separately to each objective S_UC_RHS UCEXT1 N 2 1 0 1 S_UC_RHS UCEXT2 N 2 2 0 1 S_UC_RHS UCEXT3 N 2 3 z0 1 S_UC_RHS UCEXT4 N 2 4 z0 1 S_UC_RHS UCEXTS N 2 5 z0 1 Example 3 Tradeoff curve analysis using uncertain damage costs Possible use 3 above The two phase tradeoff facility is not necessary here but a deterministic sensitivity analysis is sufficient Assume that the externality is represented by commodity COMEXT e Activate sensitivity mode and define number of analysis points SET SENSIS YES SW_SUBS 1 1 5 e Define different damage costs to be analyzed assumed constant over T S_DAM_COST R T COMEXT CUR 2 1 20 S DAM COST R T COMEXT CUR 27 27 40 S DAM COST R T COMEXT CUR 2 3 60 S DAM COST R T COMEXT CUR 2 4 80 S DAM COST R T COMEXT CUR 2 5 100 This Example 3 will also define a tradeoff curve between the externality and system costs using different assumptions for the marginal damage costs Example 4 Tradeoff analysis optimizing separately with several different objectives in the fir
16. Two alternative candidates for the objective function are e Expected utility criterion with linearized risk aversion e Minimax Regret Savage criterion Loulou and Kanudia 1999 Expected Utility Criterion with risk aversion The first alternative has been implemented into the stochastic version of TIMES This provides a feature for taking into account that a decision maker may be risk averse by defining a new utility function to replace the expected cost The approach is based on the classical E V model an abbreviation for Expected Value Variance In the E V approach it is assumed that the variance of the cost is an acceptable measure of the risk attached to a strategy in the presence of uncertainty The variance of the cost of a given strategy k is computed as follows Var C X pj Cost ECL J where COsfj is the cost when strategy k is followed and the j state of nature prevails and EC is the expected cost of strategy k defined as usual by EC X p Cost iy J An E V approach would thus replace the expected cost criterion by the following utility function to minimize U EC A 4 Var C where 4 0 is a measure of the risk aversion of the decision maker For A 0 the usual expected cost criterion is obtained Larger values of 4 indicate increasing risk aversion For 420 one gets the simple expected cost criterion Loulou R and A Kanudia 1999 Minimax Regret Strategies for Greenhouse Gas Abateme
17. all S OW In the the two phase tradeoff analysis the results are included for all S OW in the second phase In the multiphase analysis the results are reported for the single S OW in each phase unless the user explicitly turns out the reporting for certain S OW by setting a negative value for S UCOBJ 0BJ 1 SOW However results for any terminal S OW are always reported Concerning the basic results from the Climate Module the result parameter includes an index item for the various result quantities and this index is translated to the commodity dimension in VEDA BE The basic result attributes of the Climate Module are listed in Table 12 together with the standard name of the corresponding variable or result parameter in the module Since all these attributes and variables are only created by the model when the stochastic extension is used a special vdd file called times2veda stc vdd has to be applied when transferring results from TIMES to VEDA BE by using the gdx2veda utility Table 13 Basic Climate Module reporting items for stochastic TIMES Climate module Item name variable parameter Description C02 GTC VAR CLI TOT C02 GTC Total CO emissions by milestone year GtC CO2 ATM VAR CLI BOX CO2 ATM Mass of CO2 in the atmosphere GtC CO2 UP VAR CLI BOX CO2 UP Mass of CO2 in the upper ocean layer GtC C02 LO VAR CLI BOX C02 LO Mass of CO2 in the deep ocean layer GtC DELTA FORC C M DT FORC Incr
18. all those branches of the event tree that are distinct already at the first stage will be solved independently of each other If all stages are combined the stochastic scenarios will be run fully independently The SENSIS setting described above will accomplish this without the need for setting SW START Solving a set of deterministic scenarios in this way can be very useful for the following different purposes e For comparing the results from the stochastic model to the results of individual deterministic scenarios e For making standard sensitivity analysis with different values for the uncertain parameters When the model generator detects that all scenarios are disjoint already at the first stage it uses the straightforward scenano decomposition approach to solving the problem Table 11 Initial set of uncertain input parameters for stochastic TIMES Parameter Description Type S COM PROUJ r y c j w Demand projection Rel S CAP BND r y p l j w Bound on total installed capacity Abs S COM CUMPRDwr r y y c ljw Cumulative bound on commodity production Abs S COM CUMNET r y y c lj w Cumulative bound on commodity net prod Abs S_FLO_CUM r p c y y l j w Cumulative bound on flow or activity Abs S FLO _FUNC r y p cg1 cg2 j w Process transformation efficiency Rel S_NCAP_COST r y p j w Process investment cost Rel S UC RHSxxx lj w RHS constant of user constraint Abs S DAM COST r y c cur j w Damage cost of
19. aps the most important for tradeoff analyses e Uncertain RHS constants of user constraints e Uncertain damage costs Table 6 New input parameters for two phase tradeoff analysis in TIMES Parameter Description Weight coefficients for the components of the objective S UCOBJ function in the first phase of the tradeoff facility and for uc n w each SOW to be analyzed max 64 different cases Interpolation Not available Flag indicating that discounting is to be applied to the periods in the LHS side of UC constraint Applicable to UC components grp UC ACT UC FLO UC IRE UC COMPRD UC COMNET and UC COMCON for any UC summed over periods UC ATTR r uc_n LHS grp PERDISC The weight parameter W which defines the coefficients for the user defined objective components see Section 3 can be specified by using the parameter S UCOBJ as described in Table 6 Optional discounting of any flow based UC components can be activated by using the UC ATTR option PERDI SC The two phase solution procedure can be run over a maximum of 64 different cases SOWs each of which may have different values for any of the uncertain parameters The deviation bounds to be defined in Phase 2 can be specified with the UC RHSxXxx attributes by using the N bound type Any non negative N value will be applied as a deviation bound in the second phase The bound value represents the maximum proportional deviation allowed in th
20. c programming 4 3 1 User control parameters All control parameters for stochastic programming are available in the VEDA FE shell where they may be specified by the user All control parameter have a prefix SW_ in the GAMS code of the model generator The parameters are discussed in more detail below l The parameter SW START is used to indicate when each of the stochastic stages begins For stage 1 the value SW START is always assumed to be the first MILESTONYR If any SW START for subsequent stages is not equal to one of the milestone years it will be replaced by the first MILESTONYR following it If SW START is not specified for some stage gt 1 the MILESTONYR following the SW START of the previous stage is assumed In addition stages can also be combined see section 3 3 2 The parameter SW SUBS specifies the number of sub states of the world for each SOW at stage j If it is not specified for stage 1 the number is determined by using the following two rules e IfSW SUBS or SW SPROB is specified for any SOW at any stage the largest ordinal number of the SOWs in stage 2 for which either SW SUBS or SW SPROB is specified is used or 1 if none is specified at stage 2 e If neither SW SUBS nor SW SPROB is specified for any stage the largest ordinal number of the SOWs for which SW PROB is specified is used The parameter SW SPROB can be used to specify the conditional probabilities of the sub states of the world of each SOW a
21. cy Annualized undiscounted investment costs Annualized undiscounted investm taxes and subsidies Annualized undiscounted decommissioning costs Undiscounted fixed costs Undiscounted fixed taxes and subsidies Undiscounted activity costs Undiscounted flow costs Undiscounted flow taxes and subsidies Undiscounted commodity costs Undiscounted commodity taxes and subsidies Level parameters SF IN w r v t 0 C S SF OUT w r v t p c s SPAR ACTL w r v t p s SPAR_CAPL w r t p SPAR_NCAPL w r t p SPAR_COMPRDL w r t c s SPAR_COMNETL w r t c s SPAR UCSL w uc Flows into processes Flows out of processes Activity levels of processes Total installed capacities of processes Newly installed capacities of processes by period Commodity gross production levels Commodity net production levels Levels for the user constraint equations slacks Marginal parameters SPAR ACTM w r v t p s SPAR_CAPM w r t p SPAR_NCAPM w r t p SPAR_COMPRDM w r t c s SPAR_COMNETM w r t c s SPAR_COMBALEM w r t c s SPAR COMBALOGNM w r t C s Marginals for the activity variables Marginals for the total installed capacity variables Marginals for the new capacity variables Marginals for the commodity production variables Marginals for the commodity net variables Marginals for the commodity balance equations E Marginals for the commodity balance equations G Table 5 Reporting parameters for stochastic TIMES
22. e value of the LHS expression of the UC constraint as described in Section 3 Negative N bounds are ignored and therefore negative bound values can always be safely used for generating non constraining user defined equations for reporting purposes By using the uncertain S UC RHSxXxx attributes the deviation bounds to be applied in Phase 2 can be varied over S OWs The predefined UC names OBJZ and OBJI can be used in deviation bounds to refer to the original or user defined objective functions respectively OBJZ also by region OBJZ can naturally also be used also in _UCOB Remark If the objective in Phase 1 is defined for only a single SOW 1 the same objec tive will also be used for any subsequent S OW points to be analyzed according to the number of S OW as defined by SW SUBS 1 1 4 5 Stochastic variables As noted earlier the variables that are used to model the stochastic programming version of TIMES are the same variables that make up the deterministic TIMES model with two minor adjustments The main difference is that the variables require another index corresponding to the state of the world S OW To standardize the handling of this index it is always introduced after the period index thus it is usually the second index or the first if there is no period index in the variable To accommodate this requirement each standard model variable name is adjusted by replacing the standard prefix of the variable name VAR
23. e 12 Reporting parameters for stochastic TIMES Parameter Description Marginal parameters SPAR ACTM w r v t p s Marginals for the activity variables SPAR_CAPM w r t p Marginals for the total installed capacity variables SPAR NCAPM w r t p Marginals for the new capacity variables SPAR COMPRDM w rt c s Marginals for the commodity production variables SPAR_COMNETM w r t c s Marginals for the commodity net variables SPAR COMBALEM w r t c s Marginals for the commodity balance equations E SPAR COMBALGM w r t c s Marginals for the commodity balance equations G SPAR_PEAKM w r t cg s Marginals for the peak equations SPAR_UCSM w uc Marginals for the user constraint equations slacks Capacity bound parameters SPAR_CAPUP w r t p Upper bound on overall capacity in a period SPAR_CAPLO w r t p Lower bound on overall capacity in a period Climate module result parameters CM SRESULT w item t Basic results from stochastic Climate Module CM SMAXC M w y Shadow price of climate variable constraint Under the stochastic mode user constraints are modeled using slack variables with no loss in generality Therefore the reporting parameter PAR_UCSL contains the levels of the slack variables and the parameter SPAR_ UCSM represents the marginals of the slack variables undiscounted when the constraint is region specific In stochastic and sensitivity analyses the full results are produced for
24. ease in radiative forcing W m DELTA ATM C M DT TATM Increase in atmospheric temperature 9C DELTA LO IC M DT TLOW Increase in deep ocean temperature C 6 REFERENCES Dantzig G B 1955 Linear programming under uncertainty Management Science 1 197 206 Kalvelagen E 2003 Two stage stochastic linear programming with GAMS GAMS Corporation Kanudia A amp Loulou R 1998 Robust Responses to Climate Change via Stochastic MARKAL the case of Qu bec European Journal of Operations Research vol 106 pp 15 30 de Kruijk H amp Goldstein G 1999 MARKAL MUSS User Information Note Stochastic MARKAL January 20 1999 Loulou R amp Kanudia A 1999 Minimax Regret Strategies for Greenhouse Gas Abate ment Methodology and Application Operations Research Letters 25 219 230 Loulou R Goldstein G amp Noble K 2004 Documentation for the MARKAL Family of Models October 2004 http www etsap org documentation asp Loulou R Remme U Kanudia A Lehtil A amp Goldstein G 2005 Documentation for the TIMES Model Energy Technology Systems Ananlysis Programme ETSAP April 2005 http www etsap org documentation asp Wets R J B 1989 Stochastic programming In G L Nemhauser A H G Rinnoy Kan and M J Todd editors Handbook on Operations Research and Management Science volume 1 pages 573 629 North Holland 1989 Appendix A Control parameters for Stochastic TIMES
25. ey override the corresponding deterministic parameters in the parts of the event tree where they apply Absolute parameters defined at a later stage of the event tree also over ride those defined at an earlier stage All uncertain bound attributes are of absolute type Relative parameters are applied as multipliers to the corresponding deterministic base line parameters Relative parameters are also applied cumulatively over stages so that any relative parameters defined at an earlier stage of the event tree are included in the combined multipliers at a later stage Consequently for any branches downstream in the event tree the current branch represents the baseline for which the multipliers in the succeeding stage will be applied Uncertain demand projections have been implemented in TIMES as relative parameters This means that the uncertain demands are expressed as multipliers applied to the baseline demand projection The advantage of the relative parameters is that when appropriate they are easier to maintain that absolute parameters The user can also utilize the cloning facility described above for the specification of the uncertain parameters As with the stage wise conditional probabilities cloning of uncertain parameters is done at some stage j if all of the following three conditions hold The parameter has been specified for the sub states of the first S OW at stage The number of sub states was left unspecified for some other S OW at
26. for Figure 1 we have S 1995 1 S 2000 1 S 2005 1 S 2010 1 2 S 2015 1 2 3 4 2020 1 2 3 4 S 2025 1 2 3 4 S 2030 1 2 3 4 S 2035 1 2 3 4 S T set of state indices at the last stage the set of scenarios Set S T is homeomorphic to the set of paths from period 1 to last period in the event tree g ts a unique mapping from t s lseS T to S f according to the event tree g t s is the state at period corresponding to scenario s X ts the column vector of decision variables in period t under state s C t s the cost row vector p ts event probabilities A ts the LP sub matrix of single period constraints in time period t under state s b ts the right hand side column vector single period constraints in time period t under state s D ts the LP sub matrix of multi period constraints under state s e s the right hand side column vector multi period constraints under Scenario s Alternate formulation The above formulation makes it a somewhat difficult to retrieve the strategies attached to the various scenarios Moreover the actual writing of the cumulative constraints 3 is a bit delicate An alternate but equivalent formulation consists in defining one scenario per path from initial to terminal period and to define distinct variables X t s for each scenario and each time period For instance in this alternate formulation of the example there would be four variables X t
27. he event tree the user can optionally utilize a cloning facility for both the specification of the event tree and the specification of uncertain parameters At each stage cloning can be used for those S OWs for which the number of sub states of the world is equal to the number of sub states for the first SOW Cloning can in this case be activated by leaving the number of sub states unspecified for the S OWs to be cloned The model generator will then assume the same number of sub states as for the first SOW For such cloned nodes of the event tree both the conditional probabilities and the uncertain parameters of the sub states will be copied from the sub states of the first S OW whenever they have not been specified by the user The user can thus always override the cloning by simply specifying the probabilities and or uncertain parameters explicitly Cloning of the event tree can be convenient if the event tree is large because then it can considerably reduce the amount of input data needed 4 3 4 Uncertain input parameters In this first version of the stochastic TIMES the only a few uncertain input parameters have been implemented as shown in Table 2 At a later stage more uncertain parameters may of course be added All the uncertain input parameters have a prefix S_ The uncertain parameters can be divided into two types absolute and relative Absolute parameters are applied in the same way as their deterministic counterparts and th
28. he parameter SW START to indicate when each of the stochastic stages begins SW START is optional The following rules apply e For stage 1 no value needs to be specified unless some stages are combined into the first stage because the first stage always starts in the first period e If any SW START specified for subsequent stages is not equal to one of the milestone years it will be automatically replaced by the first milestone year following it e If SW START is not specified for some stage the milestone year following the SW START of the previous stage is assumed e Stages can be combined by specifying equal or decreasing values of SW START for successive stages e Equivalent deterministic runs for each scenario can be made by specifying for the first stage a value of SW START larger than any milestone year 2 Use the parameter SW SUBS to specify the number of sub states of the world for each SOW at each stage j if any The use of SW SUBS is required if more than two stochastic stages are modeled The following rules apply e If SW SUBS is not specified for stage 1 the number of states in stage 2 is determined by the model generator from the other control parameters for details see section 4 3 1 e For any subsequent stages that have sub states SW SUBS must be specified for at least the first SOW For those SOW that SW SUBS is left unspecified the number of the first SOW is assumed and these SOWs will be subject to cloning
29. indicates that set up D is selected and any non zero value indicates that set up E is to be used In addition the use of set up D requires that no other uncertain parameters apart from S UCOBJ and S UC RHSXxXxx are specified and deviation bounds can be specified only for OB 1 In the multiphase set up E all uncertain attributes can be freely used However in the multiphase case the deviation bounds in each phase can be only be specified by using the uncertain S UC RHSxxx parameters The deviation bounds defined for each OBJ 1 are in the multiphase case always preserved over all subsequent phases S OW i 1 N Any other deviation bounds defined for S OW i are also preserved unless explicitly canceled in any subsequent phase S OW k k gt i by using the N bound type in which case the bounds remain in force in phases OW i 1 k 5 6 2 Possible uses of the tradeoff facility The simple facility described above can be used for a number of tradeoff analysis tasks Below just a few possible set ups are briefly described 1 One can optimize the problem with respect to a generic N row phase 1 and then relax the optimal value of this auxiliary OBJ by n and go back to the solution of an economic model by re optimizing with the original objective function i e the total discounted costs or surplus phase 2 by iterating over N discrete values of increasing n one can build full tradeoff curves and calculate the supply cost c
30. is no need to specify any other stochastic control parameters than the number of analysis points which can be done by specifying the number of SOW SW_SUBS 1 1 N Therefore the minimal specifications required to use the two phase tradeoff analysis are the following e The stochastic mode is activated SSET STAGES SENSIS YES e The weight parameter S UCOB is defined for some UC and for either a single or several S OW 1 N see Section 3 for syntax e Ifthe analysis is to be carried over several SOW the user should additionally specify the number N of the S OWs explicitly by setting SW SUBS 1 17 N default 1 In addition the user can define any of the uncertain parameters over the SOW to be analyzed and define deviation bounds by using either the deterministic or uncertain UC RHSxxx attributes using the N bound type and a non negative bound value The predefined UC names OBJZ and OBJI can be used in the RHS parameters to refer to the original or user defined objective functions respectively OBJZ also by region The set up B described in the Introduction corresponds to the case where S UCOBJ is defined only for the first S OW The corresponding set ups D and E which both have only a single SOW in the final phase can only be accomplished by activating the stochastic mode by SET STAGES YES and by indicating the single terminal SOW by setting S UCOBJ 0BJ 1 1 where an explicit zero value EPS
31. lculation of annual costs for stochastic results Declaration of model equations parametrization for stochastics CAL VAR mod Helper for inclusion of stochastic variables in inter period equations RPT_PAR cli Calculation of the reporting parameters in the Climate Module RPT_STC cli Driver for reporting of stochastic runs in the Climate Module purpose routine for annual costs was implemented and it is used for generating the reporting cost parameters under stochastics COST ANN rpt To assist future changes in the code a small helper routine was implemented for the inclusion of generalized variables in model equations CAL VAR mod This helper routine automates the parametrization of the variables so that they will be correctly dimensioned and mapped if the model is run under stochastics The TIMES code files that were most substantially changed during the implementation are listed in Table 10 All files that involve dynamic equations are here classified to have undergone substantial changes because dynamic equations require special handling of the stochastic variables In addition the file EQMAIN mod was divided into two parts during the implementation EQDECLR mod and EQMAIN mod Moreover some files related to the ETL extension were renamed to conform to the standard conventions for TIMES extensions Table 10 Code files of the TIMES model generator with substantial changes File Description BND SET mod Uncertain bou
32. mension of the parameters The stochastic reporting parameters provide almost the same set of results as those that have been transferred to VEDA BE from standard TIMES model runs but now for each of the stochastic scenarios However there are a few small differences e All the undiscounted cost results from stochastic runs represent annualized costs and they are divided into genuine costs and taxes subsidies Decommissioning costs are annualized over the same years as fixed costs e The activity costs and flow costs are reported at the ANNUAL level only while in the standard reports they are reported in each timeslice e Reporting parameters for the levels of commodity balance and peak equations have been omitted from the stochastic reports because the levels are normally zero anyway except for demands Only the marginals are thus reported e Under the stochastic mode all user constraints are formulated by using slacks and therefore the reporting parameters for user constraints represent the levels and marginals of these slack variables Table 5 Reporting parameters for stochastic TIMES Parameter Description Cost parameters SREG WOBuJ w r item cur SCST_INVC w r v t p SCST_INVX w r v t p SCST_DECCv w r v t p SCST_FIXC w r v t p SCST FIXX w r v t p SCST ACTC w r v t p SCST_FLOC w r v t p c SCST_FLOX w r v t p c SCST_COMC w r t c SCST_COMX w r t c Discounted objective value by region type and curren
33. mo mem Energy Technology Systems Analysis Programme TIMES Version 3 0 User Note Stochastic Programming and Tradeoff Analysis in TIMES Authors Richard Loulou KANLO Consultants France Antti Lehtila VTT Finland Updated April 2011 Foreword This report contains the full documentation on the implementation and use of the Stochastic Programming and Tradeoff Analysis facilities of the TIMES model generator The report is divided in five chapters After the general introduction in Chapter 1 Chapter 2 presents a brief description of the mathematical approach taken with respect to stochastic programming and Chapter 3 the approach used for tradeoff analysis Chapter 4 contains the description of the GAMS implementation of the new elements along with the sets parameters variables and equations that have been added to the TIMES model Finally Chapter 5 summarizes the usage notes in the form of a brief User s Manual for stochastic programming and tradeoff analysis in TIMES This documentation may eventually also be inserted in the complete documentation of the TIMES model Table of contents DENM INTRODUC TO N eere TAE 5 2 MULTI STAGE STOCHASTIC PROGRAMMING ee 7 2 1 Gelieral cerechiuranivvcececuvanetvasancoancuwhusiacisccsbcldacusensescetancaaceastonesscetenmntcueaeassunlaaterancartonsvees 7 22 Alternative objective formulationsS sso sesesesesososcsosossesosesesosososososossoesosososss
34. model is sequentially run over the set of SOWs using the corresponding values of the uncertain attributes in each individual run Figure 2 illustrates a few possible set ups for sensitivity and tradeoff analyses in TIMES all of which are supported by the model generator A Simple sensitivity analysis over the set of SOWs B Two phase tradeoff analysis where the model is first run once using a user defined objective function and then the solution from the first phase is used for defining additional constraints in a series of model runs in the second phase C Two phase tradeoff analysis where the model is first run over a set of SOWs each of which may have a different objective functions and different parameter attributes In Phase 2 the solution for each SOW from Phase 1 is used for defining additional constraints for each SOW in Phase 2 where the standard objective function is used D Two phase tradeoff analysis where the model is first run over a set of SOWs each of which have a different objective function and optionally different UC RHS In Phase 2 the solution for each SOW obtained from Phase 1 is used for defining an additional deviation constraint for each of the objectives used in Phase 1 and a single model is solved in Phase 2 optimizing the standard objective function E Multiphase tradeoff analysis over N phases with different objective functions Sensitivity Two phase tradeoff analysis Multiphase analysis B C D
35. nd parameters for capacities BNDMAIN mod Handling of stochastic indexes for bounds CAL CAP mod Dynamic equations for capacity related flows CAL NCOM mod Dynamic equations for investment and decommissioning flows EQCAPACT mod Dynamic capacity utilization equations EQCOMBAL mod Uncertain demand parameters EQCPT mod Dynamic capacity transfer equations EQCUMCOM mod EQDAMAGE mod EQMAIN mod EQOBJ mod EQOBJELS mod EQOBJFIX mod EQOBJINV mod EQOBJVAR mod EQOBSALV mod EQPEAK mod EQSTGIPS mod EQUSERCO mod MAINDRV mod MOD EQUA mod RPTMAIN mod ATLEARN etl EQU EXT etl EQCAFLAC vda EQU EXT cli RPT EXT cli Dynamic cumulative commodity equations Dynamic equations for objective component of damages Handling of basic equation differences under stochastic TIMES Objective functions for stochastic programming Dynamic equations for objective component Dynamic equations for objective component Dynamic equations for objective component Dynamic equations for objective component Dynamic equations for objective component Dynamic capacity peaking equations Dynamic storage equations Dynamic and cumulative user constraints Handling of the main control variables for stochastic TIMES Parametrized equation declarations Handling of report generation under stochastic TIMES Reports from the ETL extension under stochastic TIMES Dynamic learning equations former EQUETL etl Dynamic capacity utilization equations Dynamic carbon balance equatio
36. net production of commodity Abs S_CM_MAXC y item j w Bound on maximum level of climate variable Abs S CM CONST item j w Climate module constant CS or SIGMA1 Abs Thus each scenario is in this case solved separately one after another The results are still reported in the same way as in a standard stochastic run 5 5 Example Five stage stochastic model In section 4 3 4 a simple example of specifying a stochastic model was already given In this section a somewhat larger and more complete example is given which however uses the same uncertain parameters as the earlier example In the run file the stochastic mode should be activated as follows e as stochastic S The specification of the event tree consists of the definition of the starting years of the stages the number of sub states of each sow and the probabilities We are defining a five stage event tree illustrated in Figure 3 In this example we define the second stage to start in 2010 and the third stage in 2020 We leave the start years of the fourth and fifth stage unspecified which means that they start in the milestone year succeeding the previous stage default definition PARAMETER SW START 2 2010 3 2020 We wish to have three states at the second stage and all states will have two sub states at each subsequent stage Here we can utilize the cloning facility and therefore we need to specify the number of sub states for the first state at each s
37. ns dynamic concentration bounds Handling of report generation under stochastic TIMES 5 USER S REFERENCE 5 1 Activating the stochastic mode The stochastic mode can be activated by using the following setting in the run file SET STAGES TES If the stochastic mode is used for sensitivity analysis only it can be alternatively acti vated also by using the following setting SET SENSIS YES All the control and input parameters of the stochastic extension are only available when using either of these settings When using the SENSI S setting for sensitivity analysis the model will be solved sequentially in each of the stochastic scenarios using the basis information from each run as a starting basis for the next run 5 2 Specification of states of the world and scenarios The user does not need to specify the stages states of the world or scenarios explicitly The predefined domain for the stages is the set J which contains the elements 1 2 50 This should be sufficient for any conceivable stochastic TIMES model The predefined domain for the states is the set ALLSOW which contains the elements 1 2 64 Consequently a maximum of 64 states at any given stage can be used in the specification of a stochastic model The same maximum amount applies of course also to the scenarios which are the states of the last stage 5 3 Specification of input parameters 5 3 1 Specification of control parameters 1 Use t
38. nt Methodology and Application Operations Research Letters 25 219 230 Taking risk aversion into account by this formulation would lead to a non linear non convex model with all its ensuing computational restrictions These would impose serious limitations on model size Utility Function with Linearized Risk Aversion To avoid non linearities it is possible to replace the semi variance by the Upper absolute deviation defined by UpAbsDe Cost Cost EG i j where y x is defined by the following two linear constraints y 2x and y gt 0 and the utility is now written via the following linear expression U EC A UpsAbsDewC This is the expected utility formulation implemented into the TIMES model generator Note that this linearized version of the risk averse utility function is not available in the MARKAL code 2 3 Solving approaches General multi stage stochastic programming problems of the type described above can be solved by standard deterministic algorithms by solving the deterministic equivalent of the stochastic model This is the most straightforward approach which may be applied to all problem instances However the resulting deterministic problem may become very large and thus difficult to solve especially if integer variables are introduced but also in the case of linear models with a large number of stochastic scenarios Two stage stochastic programming problems can also be solved efficie
39. ntly by using a Benders decomposition algorithm Kalvelagen 2003 Therefore the classical decompo sition approach to solving large multi stage stochastic linear programs has been nested Benders decomposition However a multi stage stochastic program with integer vari ables does not in general allow a nested Benders decomposition Consequently more complex decompositions approaches are needed in the general case e g Dantzig Wolfe decomposition with dynamic column generation or stochastic decomposition methods The current version of the TIMES implementation for stochastic programming is solely based on directly solving the equivalent deterministic problem As this may lead to very large problem instances stochastic TIMES models are in practice limited to a relatively small number of scenarios 3 TRADEOFF ANALYSIS Analyzing tradeoffs between the standard objective function and some other possible objectives for which the market is not able to give a price has not been possible in an effective way until TIMES version 2 5 0 The tradeoff analysis facility is available under the stochastic mode of TIMES which provides the basic tool for making sensitivity analyses over a number of different cases In addition to providing the means for specifying the parameters to be varied in the sensitivity analysis the tradeoff facility provides a tool for making a two phase or a multiphase tradeoff analysis using a completely user defined objective f
40. ntrol parameters for Stochastic TIMES APPENDIX B Input parameters for Stochastic TIMES l NTRODUCTION Stochastic Programming is a method for making optimal decisions under risk The risk consists of uncertainty regarding the values of some or all of the LP parameters cost coefficients matrix coefficients RHSs Each uncertain parameter is considered to be a random variable usually with a discrete known probability distribution The objective function thus becomes also a random variable and a criterion must be chosen in order to make the optimization possible Such a criterion may be expected cost expected utility etc as mentioned by Kanudia and Loulou 1998 Uncertainty on a given parameter is said to be resolved either fully or partially at the resolution time i e the time at which the actual value of the parameter is revealed Different parameters may have different times of resolution Both the resolution times and the probability distributions of the parameters may be represented on an event tree such as the one of figure l depicting a typical energy environmental situation In figure 1 two parameters are uncertain mitigation level and demand growth rate The first may have only two values High and Low and becomes known in 2005 The second also may have two values High and Low and becomes known in 2010 The probabilities of the outcomes are shown along the branches This example assumes that present time is 1995 This
41. on in TIMES i e the total discounted system costs In addition in the second phase the user can specify bounds on the proportional deviation in the LHS value of any user constraint in comparison to the optimal LHS value obtained in the first phase Such deviation bounds can be set for both global and non global constraints and for both non constraining and constrained UCs any original absolute bounds are overridden by the deviation bounds The objective function used in Phase 1 is now available as an additional pre defined UC named OBJ1 so that one can set either deviation bounds or absolute bounds on that as well if desired In addition both the total and regional original objective function can be referred to by using the pre defined UC name OBJZ in the deviation bound parameters The objective function to be minimized in the second phase and the additional bounds on the LHS values of UCs can be written as follows minobjz LHS OBJZ LHS uc 1 4 maxdev uc LHS uc for each uc for which LHS uc 2 1 maxdev uc LHS uc maxdev uc has been specified where LHS OBJZ the standard objective function discounted total system costs LHS uc LHS expression of user constraint uc according to its definition LHS uc optimal LHS value of user constraint uc in Phase 1 maxdev uc user specified fraction defining the max proportional deviation in the value of LHS uc compared to the
42. option PARAMETER S CM MAXCO2C 0 2 1 1 2010 2 1 900 2080 2 1 900 0 2 2 1 2010 2 2 900 2080 2 2 1150 0 2 3 1 2010 2 3 900 2080 2 3 1400 The uncertainty concerning demand is assumed to be resolved gradually at the sub sequent stages At each stage we assume a high and low growth scenario At the third stage high growth is 7 higher and low growth is 4 lower than the baseline At the fourth stage high growth is 5 higher and low growth is 5 lower than the adjusted baseline scenarios at previous stage Finally at the fifth stage high growth is 4 higher and low growth is 7 lower than the adjusted baseline scenarios at the previous stage The cloning facility can again be effectively utilized for the demand data Instead of specifying the demand scenarios for all of the 42 nodes at the third fourth and fifth stage we can specify the data for only six nodes corresponding to the sub states of the first nodes at the second third and fourth stage Note that the demand parameters are by O O Stagel Stage2 Stage3 Stage4 Stage5 2000 2010 2020 2030 2040 2050 2060 2070 2080 Figure 3 Event Tree for a five stage stochastic TIMES example default interpolated and extrapolated over all valid periods for each stage and thus only one data point is needed The demand scenarios can thus be fully specified as follows PARAMETER 5 COM PROJ REG 2015 DEM 3 1 1 07 REG 2015 DEM 3 2 0 96 REG 2015
43. pes or Lambda values Assume that the generic externality is represented by UCEXT which should be minimized Activate stochastic mode and define number of tradeoff points SET STAGES YES SW_SUBS 1 1 5 Define composite objective functions to be minimized in Phase 1 S UCOBJ OBJZ 1 2 1 S_UCOBJ UCEXT 1 Lambdal S_UCOBJ OBJZ 2 1 S_UCOBJ UCEXT 2 Lambda2 S_UCOBJ OBJZ 3 1 S_UCOBJ UCEXT 3 Lambda3 S_UCOBJ OBJZ 4 1 S_UCOBJ UCEXT 4 Lambda4 S_UCOBJ OBJZ 5 1 S_UCOBJ UCEXT 5 Lambda5 Define deviation bounds to be applied in Phase 2 Here as an example we wish to retain the optimal value of the composite objective obtained in Phase 1 while minimizing the total system costs in Phase 2 UC_RHS OBJ1 N EPS Example 2 Two phase tradeoff equivalence space analysis Possible use 2 above with iteration over 5 different objectives Assume that the generic rows to be optimized in Phase 1 are UCEXTI UCEXT2 UCEXT3 UCEXT4 and UCEXTS of which the even numbered should be maximized Activate stochastic mode and define number of tradeoff points SET STAGES YES SW_SUBS 1 1 5 Define an upper bound for OBJZ in Phase 1 UC_RHS OBJZ UP 7e6 Define objective functions to be used in Phase 1 S_UCOBJ UCEXT1 1 1 S_UC
44. re the same equations that make up the non stochastic TIMES model with two minor adjustments The main difference is that the equations require another index corresponding to the state of the world S OW To standardize the handling of this index either the SOW index as such or the set SW T tsow is introduced after all the other indexes of the equation The set SW T is used instead of SOW whenever the equation concems a single period To accommodate the required modifications each standard model equation name is adjusted by replacing the standard prefix in the equation name EQ by ES So for example the capacity transfer equation EQ CPT r p becomes ES CPT r Lp SW T tsow During matrix generation the appropriate S OW index value is then entered into the ES_ CPT equations according to the set SW T and the period being worked on As there is thus essentially no redefinition of the equations for the stochastic formulation other than the objective function below and the control over the generation of the appropriate equations and variables according to the control sets SW T and SW TSW mentioned above the user is referred to Chapter 5 of the TIMES Reference Manual for details on the core equations of the model Below in Table 8 the few equations directly related to only stochastic version are listed and briefly described The equations include the standard expected value for the stochastic objective function the deviation equations and finall
45. results to VEDA BE For reporting the results of the stochastic models the attributes listed in Table 12 have been added for the transfer of results into VEDA BE Table 12 Reporting parameters for stochastic TIMES Parameter Description Cost parameters SREG WOBuJ w r item cur SCST_INVC w r v t p SCST_INVX w r v t p SCST_DECCv w r v t p SCST_FIXC w r v t p SCST FIXX w r v t p SCST_ACTC w r v t p SCST_FLOC w r v t p c SCST_FLOX w r v t p c SCST_COMC w r t c SCST_COMX w r t c Discounted objective value by region type and currency Annualized undiscounted investment costs Annualized undiscounted investm taxes and subsidies Annualized undiscounted decommissioning costs Undiscounted fixed costs Undiscounted fixed taxes and subsidies Undiscounted activity costs Undiscounted flow costs Undiscounted flow taxes and subsidies Undiscounted commodity costs Undiscounted commodity taxes and subsidies Level parameters SF IN w r v t 0 C S SF OUT w r v t p c s SPAR ACTL w r v t p s SPAR CAPL w r t p SPAR_NCAPL w r t p SPAR_COMPRDL w r t c s SPAR_COMNETL w r t c s SPAR_UCSL w uc Flows into processes Flows out of processes Activity levels of processes Total installed capacities of processes Newly installed capacities of processes by period Commodity gross production levels Commodity net production levels Levels for the user constraint equations slacks Tabl
46. riginal objective function be used in the last phase so that the economic sense is maintained in the final solution 4 GAMS IMPLEMENTATION 4 1 Overview The handling of multi stage stochastic programming has been implemented into the GAMS code of the TIMES model generator The stochastic mode is activated by the following setting in the run file SET STAGES YES All the required control and input data parameters must also be specified as explained in the following sections The stochastic results can be made available to the VEDA BE report generator as explained in Section 5 6 4 2 Stages states of the world and scenarios The predefined set J constitutes the domain of the stochastic stages The members of this predefined set are named 1 2 3 50 Therefore in principle a maximum of 50 stages could be defined in the event tree The actual number of stages in a model will be one larger than the sequence number of the last stage for which the number of sub states SW SUBS is specified see below The predefined set ALLSOW constitutes the domain of possible states of the world Currently it has been defined to include the members 1 2 3 64 In other words the maximum number of states of the world is 64 Consequently a binary event tree could include at most 7 stages because 2 6 64 In each stage of the event tree the states of the world are identified by sequential integers starting from 1 For example if
47. s being used in the context of this parameter as well as internal parameters sets or result parameters being derived from the input parameter This column lists the unit of the parameter the possible range of its numeric value in square brackets and the inter extrapolation rules that apply An indication of circumstances for which the parameter is to be provided or omitted as well as description of inheritance aggregation rules applied to parameters having the timeslice s index Equations or variables that are directly affected by the parameter Abbreviation i e inter extrapolation Appendix B Input parameters for Stochastic TIMES Input parameter Related Units Ranges amp Instances Description Affected equations or Indexes parameters Default values amp Default Required Omit Special variables inter extrapolation conditions S_COM_PROJ com_proj e Commodity unit e Only applicable to demand Multiplier for projected Affects the RHS of the r datayear c j w com fr e 0 INF commodities annual demand of a commodity balance default value none Default i e standard commodity constraint EQ COMBAL S COM CUMNET r yl y2 c bd j w s com cumprd com cumprd com cumnet rhs combal e Commodity unit 0 INF default value none Default i e NA e The years yl and y2 may be any years of the set allyear Bound on the cumulative net amount of a commodity c between
48. seceossosococscssesosocscssesoscssesoe 23 4 8 Changes in model generator Code sesossessescscsocscssesososeceossosococecssseosocscsseeososoesoe 23 5 USEBSAHBEFERENGE EE aicE UO 3025082322488 MERE Ara da EE aana aaen i icasi saisit 25 5 1 Activating the stochastic modle eee eee eee eee eee eene reete eene tne ts ea etnee 25 5 2 Specification of states of the world and scenarios eere 25 5 3 Specification of input parameters ccccccscccssssssssssesssccsccssssssesssessessesseeces 25 5 3 1 Specification of control parameters 1 ecco tete aeta or err eee erepta vette 25 5 3 2 Specification of uncertain parameters sese 26 5 4 Sensitivity ANALYSES sxcsscisiscicansidescasasisussenideieccdavcusesdscateddesssbioasstenddbatebiscceawioussscccdnaes 27 5 5 Example Five stage stochastic model eere eese ee eee eee eene enean 28 5 6 Tradeohf and Iyse8 isse actis etus rutas Dri ERR a sasana eSa Db o de i Qu aiae 30 5 6 1 Activating the tradeoff analysis mode see 30 5 6 2 Possible uses of the tradeoff facility i2 deett reete idet Eid 31 5 7 Tradeoff analysis examples sesososossoesosososscesesososocecssososocscssesososossssosoesosscsosossosoe 32 5 8 Exporting results to VEDA BE sssssesosososscesesosococscssesococscssceossosososeceossososseecsseso 35 6 idgdiz lez 38 APPENDIX A Co
49. seosse 9 2 3 Solving approdc Des ia ccssviscccessecwacurshiuasacntcaussdeunekssaraceusesvaceascusuuiaaceveacenaussuseuntaseatvace 10 3 STRADEOFE ANALYSIS 3 6vedtumdtdeiuass tiie Ada akties 11 3 1 Two phase tradeoff amalysis ssscccsssssssssscssscsccscccssssccsccsessessescessesssscsessesssses 11 32 Multiphase tradeoff analysis ccccsccsccsscssccsscscesscsscsescessessssecessessescesersers 12 4 GAMS IMPLEMENTATION sisiscececcsiscetsntssessadsecsssnvesstuciowssdecutuadasuassonadedesiens 13 4 1 dg 2 hd m NEED D T PLE DIE 13 4 2 Stages states of the world and scenarios eere eee eee eene eerte etn aene 13 4 3 Parameters for stochastic programming ces eee eee eere eene ener n etna a enano 14 4 3 User control parameters Lau ede sve toph s Reed iraa ENa iapa 14 4 3 2 COMBS Sta OOS caa refe e a aTa aE aR RIR eT Reese hs 15 4 3 3 Clo mig parts or tie event WOE se cien ded diete canada taie aaia 15 4 3 4 Uncertain PUL pardmeletsea iioc acero aiaa a e RU Bate ht 16 4 3 5 Internal sets parameters and control variables esses 18 4 3 6 Reporting parameters ooa coa esed ien ane a ER RUE RE UR Reto I RU Eiaa as 18 4 4 Parameters for tradeoff analysis eere ee reete eee eene eene en eene ena en aeta 20 4 5 Stochastic Vair Ker e 21 4 6 GUAUIONS EEE E E O A 22 4 7 Supported TIMES extensions s sssssssesosccsessescscsocscssososo
50. ser constraints r uc_n y lim j w open default value bound type of lim of EQ I UCRT none a user constraint Default i e migrate to period year S UC RHSTS uc rhsts None Used in user constraints RHS constant with User constraints uc n y s lim j w open default value none Default i e migrate to period year bound type of lim of a user constraint EQ UCTS Units Ranges amp Instances Description Affected equations or E ee Sis Default values amp Default Required Omit Special variables Indexes PREISE inter extrapolation conditions S UC RHSRTS uc rhsrts e None e Used in user constraints RHS constant with e User constraints r uc_n y s lim j w e open default value none e Default i e migrate to bound type of lim of a user constraint EQ _UCRTS period year S_UCOBJ e None Used for defining a user Weight coefficient of e Objective function uc n w e open default value defined objective function in the objective component ES EXPOBJ none Phase 1 of the tradeoff uc n in analysis point w EQ OBJ e Default i e n a analysis facility
51. solution in Phase 1 Remarks 1 Use of the two phase tradeoff analysis facility requires that a weight has been defined for at least one objective component in the first phase 2 If no deviation bounds are specified the second phase will be omitted 3 Automatic discounting of any commodity or flow based UC components is possible by using a new UC ATTR option PERDISC see Section 4 4 which could be applied e g to the user defined objective components in Phase 1 4 The two phase tradeoff analysis can be carried over a set of distinct cases each identified by a unique SOW index 3 2 Multiphase tradeoff analysis The multiphase tradeoff analysis is otherwise similar to the two phase analysis but in this case the objective function can be defined in the same way as in the Phase 1 described above also in all subsequent phases The different objective functions in each phase are distinguished by using an additional phase index the SOW index Deviation bounds can be specified in each phase such that they will be in force over all subsequent phases any user constraints or only in some of the succeeding phases any user constraints excluding OBJ1 The deviation bounds defined on any of the user defined objectives OBJI will thus always be preserved over all subsequent phases Remark 1 Although the multiphase tradeoff analysis allows the use of any user defined objective functions in each phase it is highly recommended that the o
52. st phase and using deviation bounds for each of them in a single run in the second phase Possible use 6 above Assume that the generic rows to be optimized in Phase 1 are UCEXTI UCEXT2 UCEXT3 UCEXT4 and UCEXTS of which the even numbered should be maximized e Activate stochastic mode and define number of tradeoff points SET STAGES YES SW_SUBS 1 1 5 e Define an upper bound for OBJZ in Phase 1 UC_RHS OBJZ UP 7e6 e Define objective functions to be used for each SOW S_UCOBJ UCEXT1 1 1 S_UCOBJ UCEXT2 2 2 1 S_UCOBJ UCEXT3 3 1 S_UCOBJ UCEXT4 4 2 1 S_UCOBJ UCEXTS 5 1 e Seta flag indicating that a single model is to be used in the second or last Phase Any non zero value for the flag would activate the multiphase set up E thereby optimizing UCEXTS in the last phase under deviation bounds on the other objectives However in this example the two phase set up D is desired and therefore an explicit zero value has to be specified for this flag S_UCOBJ OBJ1 1 EPS e Define deviation bounds to be applied in Phase 2 In this example 10 or 20 deviation bounds are applied to odd and even numbered objectives respectively S UC RHS OBJI N 27 1 0 1 S UC RHS OBJI N 27 2 0 2 S_UC_RHS OBJ1 N 2 3 0 1 S UC RHS OBJI N 27 4 0 2 S UC RHS OBJI N 27 5 0 1 5 8 Exporting
53. system costs without risk aversion 4 3 2 Combining stages Stages of the event tree can also be combined if deemed useful Any successive stages will be combined into a single stage if the starting year of the succeeding stage is less than or equal to the preceding stage For example if in the example of Figure 1 the starting year of both stages 2 and 3 would be specified to be 2010 the stages 2 and 3 would be combined so that in 2010 the event tree is expanded directly from one state of the world to four states By using this feature stages can even be combined with the first stage by specifying the same value of SW START for both stage 1 and some subsequent stages If all stages were combined with the first stage the resulting model would optimize all the scenarios independently of each other This feature can be used for making a deterministic run for each scenario This can be done best by specifying a large value of SW START for the first stage and by leaving the other values intact Combined stages can be also useful for data management for example when the states at stage 2 should contain some combinations of uncertain parameters In such cases it can be useful to define the scenarios for the uncertain parameters at successive stages so that cloning becomes possible Then the combined scenarios for stage 2 can be formed by combining these successive stages 4 3 3 Cloning parts of the event tree If there are more than 2 stages in t
54. t each stage for which cloning is applied SW MAP t w j w SW STAGE j w SW T tw SW TOS w tw SW TREE j w w SW_TSTG t j SW TSW w t w SW UCT uc_n t w Mapping from period and internal SOW to stage and original SOW Internal SOWs at each stage Valid internal SOWs in each period Mapping from redundant scenarios to unique SOW in each period t Scenarios for each original SOW at each stage Valid stages j for each period t Mapping from all scenarios to unique SOW in each period t Valid internal SOWs in each period for period wise user constraints Parameter Description SW DESC j w SW TPROB t w Number of scenarios for each original SOW at each stage Probability of each internal SOW in each period 4 3 5 Internal sets parameters and control variables The implementation uses a few internal sets and parameters All the internal sets and parameters have a prefix SW Table 4 gives an overview of these sets and parameters The implementation of the stochastic extension uses a number of GAMS control variables for renaming and adjusting the equations and variables for the additional dimension needed for stochastic programming Table 3 summarizes the control variables 4 3 6 Reporting parameters All standard reporting parameters have a prefix S and the first index is always the stochastic scenario index Reporting parameters for the Climate Module have a prefix CM S The scenario index is always the first di
55. t each stage j Another way to specify the probabilities is to specify the total probabilities of each SOW at the last stage see SW PROB Table 1 Control parameters for Stochastic TIMES Parameter Description The year corresponding to the resolution of uncertainty at each SW START j stage and thus the last year of the hedging phase and the point from which the event tree fans out for each of the SOW SW_SUBS j w The number of sub states of the world for each SOW at stage j SW SPROB w SW PROB w The conditional probability of each sub state at stage j These conditional probabilities can be overridden by SW PROB The total probability of each SOW at the last stage If specified overrides the stage specific conditional probabilities SW LAMBDA Risk aversion coefficient 4 The parameter SW PROB can be used to specify the total probability of each SOW at the last stage If specified the total probability will override the total probability derived from the stage wise conditional probabilities Another way to specify the probabilities is thus to specify the conditional probabilities for the sub states of the world at each stage If the resulting final total probabilities will not sum up to 1 they will be simply normalized over all SOWs 5 The parameter SW LAMBDA can be used to specify the risk aversion coefficient If not specified the objective function represents the expected total discounted
56. tage only and not for each state of each stage PARAMETER SW SUBS Poil X d 312 1320 If we wish to specify the conditional probabilities for the sub states so that the distri bution is the same under each parent state we can use the cloning facility quite effec tively for the specification of probabilities For the second stage we need to define the probabilities for the first two states the third is derived automatically For the remaining event tree we only need to define the probability for the first state at each stage the second is derived PARAMETER SW SPROB 2 1 0 33 2 2 0 34 3 1 0 60 4 1 0 55 5 1 0 5 However if we wish to override the cloning of some probabilities we can do that by specifying the probabilities explicitly Below the probabilities of the sub states of the last branch at stage 4 are specified explicitly PARAMETER SW SPROB 4 12 0 7 The uncertainty concerning climate change mitigation is assumed to be resolved at the first stage We assume a high medium and low mitigation scenarios In this example we assume that the high scenario corresponds to a CO concentration limit of 900 GtC in 2080 the medium scenario to 1150 and the low scenario to 1400 GtC in 2100 respectively For all scenarios we assume that the limit evolves linearly from the value of 900 GtC in 2010 The mitigation parameters can then be specified as follows the year index 0 is a placeholder for the interpolation control
57. tage stochastic programming also with the following TIMES extensions that are included in the standard distribution The Climate Module CLI The Lumpy Investment extension DSC The Endogenous Technological Learning extension ETL The Damage Cost Functions DAM The TIMES VEDA FE extension VDA The IER extension of University of Stuttgart IER The stochastic mode cannot be used with the new TIMES MACRO model variant 4 8 Changes in model generator code The implementation required extensive modifications to the existing code as well as a number of new components in the model generator In total about 90 existing files were modified and 11 new code files were added The new code components are listed in Table 9 The new files that are solely related to stochastic TIMES have the extension stc with the exception of RPT_STC cli which is the report driver for the Climate Module under stochastic programming A new general Table 9 New files in the TIMES model generator code File Description INITMTY stc Declarations for the stochastic extension STAGES stc Preprocessing and management of stochastic stages RPTMAIN stc Main driver for reporting results from stochastic runs FILLSOW stc Cloning and processing of uncertain input parameters RENAME stc Reporting of stochastic results in scenario files used for ETL only SOLVE stc Solving stochastic model with possible direct decomposition COST ANN rpt EQDECLR mod Ca
58. the years yl and y2 within a region Forces the variable VAR_COMPRD to be included in the balance EQE_COMBAL Generates cumulative commodity constraint EQ _CUMPRD S COM CUMPRD r yl y2 c bd j w S com cumnet com cumprd com cumnet rhs comprd e Commodity unit 0 INF default value none Default i e NA e The years yl and y2 may be any years of the set allyear Bound on the cumulative production of a commodity between the years yl and y2 within a region Forces the variable VAR_COMNET to be included in the balance EQE_COMBAL Generates cumulative commodity constraint EQ _CUMNET The first row contains the parameter name the second row contains in brackets the index domain over which the parameter is defined This column gives references to related input parameters or sets being used in the context of this parameter as well as internal parameters sets or result parameters being derived from the input parameter This column lists the unit of the parameter the possible range of its numeric value in square brackets and the inter extrapolation rules that apply An indication of circumstances for which the parameter is to be provided or omitted as well as description of inheritance aggregation rules applied to parameters having the timeslice s index Equations or variables that are directly affected by the parameter Abbreviation i e inter extrapolation
59. there are three states in the second stage these are identified by the numbers 1 2 and 3 If all these three states have two sub states in the third stage those will be numbered 1 2 6 so that the states 1 and 2 in the third stage are sub states of state 1 in the second stage The states of the world defined for the final stage of the event tree constitute the actual set of different final states to be handled also called scenarios The set of final states S OW is then of course also a subset of the domain ALLSOW The alias name W is defined for SOW and the name WW is defined for ALLS OW Internally the states of the world are numbered differently in the intermediate stages The internal numbering is obtained by enumerating all stages p excluding the last stage in reverse order If the first sub state of a certain state in stage p is k in stage p J this state will be internally numbered k in stage p instead of the sequential number However actually the user does not need to know anything about this internal numbering as all the input parameters will use the original numbering based on sequential numbers at each stage The results on the other hand are for all periods reported for all of the states at the final stage because the states at the final stage represent the unique scenarios across the periods The mapping of the scenario indexes to the original state indexes in each period is left to the user 4 3 Parameters for stochasti
60. unctions in the first phase or even in several phases 3 1 Two phase tradeoff analysis In the first phase of the TIMES two phase tradeoff analysis facility the objective function can be defined as a weighted sum of any number of objective components All of the components will refer to the LHS value of a global user constraint i e a user constraint that is summed over regions and periods Each of the component UCs can be either fully non constraining or constrained by upper lower bounds on the LHS The components are defined by the user by specifying non zero weight coefficients for the UCs to be included in the objective The original objective function total discounted costs is automatically pre defined as a non constraining user constraint with the name OBJZ and can therefore always be directly used as one of the component UCs if desired Consequently the first phase can be considered as representing a simple Utility Tradeoff Model which can also be used as a stand alone option The resulting objective function to be minimized can be written as follows minobjl W uc LHS uc uceUC GLB where W uc weight of objective component uc in Phase 1 LHS uc LHS expression of user constraint uc according to its definition UC GLB the set of all global UC constraints including OBJZ In the second phase of the TIMES two phase tradeoff analysis facility the objective function is always the original objective functi
61. urve of the public good reflected by row N 2 One can preliminarily set an upper bound to the cost objective OBJZ then as in point one above one can optimize the problem with respect to a generic N row phase 1 and then relax the optimal value of this auxiliary OBJ by n and go back to the economic equilibrium by re optimizing with the original objective function i e the total discounted costs or surplus phase 2 by iterating over N different rows it is possible to identify the equivalence space of different public goods One can use uncertain damage costs to define the tradeoffs between the externalities and the standard system costs In this way one can also introduce threshold levels for the externalities as well as non linearity in the value of the public good One can also use more complex user defined objective functions in the first phase One can use phase 1 only by specifying a linear combination of OBJZ and some other criterion By varying the weight of that criterion one may obtain a full parametric analysis of the trade offs between OBJZ and the other criterion It is easily shown using Duality theory that this is mathematically equivalent to the sensitivity analysis described in 1 above One can optionally set an upper bound to the cost objective OBJZ and optimize the problem separately with respect to a set of N different generic N rows phase 1 and then relax the optimal value of each of the auxiliary objectives
62. y the formula for the generalized objective function including the risk aversion Table 8 Equations for stochastic TIMES Equation Description ES SOBJ Nw The total discounted cost associated with each SOW A The expected value of the total discounted system cost taking ES EXPOBJ into consideration the probability of each event path E B Under the two phase Tradeoff Analysis the user defined objective function in the first Phase The upside absolute deviation between the total system cost for ES UPDEV w ind SOW and the expected value of the total system cost ese equations are generated only when the risk aversion coefficient SW LAMBDA has been specified The multi objective function i e the full objective function whether risk is accounted for or not by adding to the expected EQ OBJ cost a risk term obtained by multiplying the risk aversion intensity SW LAMBDA by the upside absolute deviation probability weighted sum of the upside deviations for every state of world penalties Note that the equation for the final objective function EQ 0B has the same name as in standard TIMES only the definition of this equation is different under the stochastic mode Similarly also the objective variable ObjZ has the same name as in standard TIMES 4 7 Supported TI MES extensions The implementation of stochastic programming has been extended to support the use of multi s

Stochastic Programming and Tradeoff Analysis in TIMES

Contents

Download Pdf Manuals

Related Search

Related Contents