ROBUST SCHEDULING UNDER TIME-SENSITIVE ELECTRICITY PRICES FOR CONTINUOUS POWER- INTENSIVE PROCESSES

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1 ROBUST SCHEDULING UNDER TIME-SENSITIVE ELECTRICITY PRICES FOR CONTINUOUS POWER- INTENSIVE PROCESSES Sumit Mitra *a, Ignacio E. Grossmann a, Jose M. Pinto b and Nikil Arora c a Carnegie Mellon University, Pittsburg, PA b Praxair Inc., Danbury, CT c Praxair Inc., Tonawanda, NY Abstract Optimization models for te production sceduling of power-intensive processes suc as air separation can elp realizing significant economical savings. However, if te electricity is procured in te dayaead or real-time market, te forecast for te ourly electricity prices contains a significant amount of uncertainty. In tis work, we apply robust optimization to te uncertain electricity prices using an uncertainty set tat features multiple ranges and can account for correlated data. We describe ow te robust counterpart is constructed and provide a case study tat sows te differences in te solutions for te deterministic and te robust scedule. Keywords Power-intensive processes, air separation plant, time-varying electricity prices, demand response, robust optimization, multiple ranges, correlated data. Introduction Te profitability of power-intensive processes, suc as air separation plants, is largely dependent on te ability to quickly adapt to canges in te electricity prices. Wit increasing volatility in te electric power market due to deregulation and an increasing sare of renewable energies, te need for a smart energy management is even more pressing. In recent work (Mitra et al., 2011), we describe an optimization model for continuous powerintensive processes tat facilitates reducing operating expenses due to electricity. Applying te metodology for weekly liquid production planning of different underutilized air separation plants sows savings of te order of 10 % and more wen compared to a simple set point euristic. In te case study, te electricity prices are assumed to be known wit certainty, wic is accurate for time-of-use (TOU) pricing were on-peak, off-peak and mid-peak prices are usually negotiated for an entire season of te year. However, for more volatile pricing scemes, suc as day-aead or real-time pricing, te electricity forecast contains uncertainty, wic we need to address. Background One common approac to deal wit parameter uncertainty is te framework of stocastic programming, wic as been applied for a medium-size air separation unit by Ierapetritou et al. (2002). However, stocastic programming problems tend to become large wit an increasing number of scenarios and are ard to solve. Furtermore, scedules are optimized for te expected total cost and usually do not explicitly protect against various realizations of te uncertain parameters. In contrast to tat, te framework of robust optimization focuses on a computationally tractable * To wom all correspondence sould be addressed

2 description of an uncertainty set, against wic te solution is robustified. Te robustness of te scedule for air separation plant requires special attention wen slow plant dynamics in terms of switcing beavior are considered. In te case of a weekly scedule for air separation plants, operational constraints suc as minimum downtime (in te order of 24 ours for larger plants) and minimum uptime (48-60 ours) can considerably reduce te degrees of freedom once a sutdown is performed. Furtermore, operating personnel favor robust scedules wit respect to anticipated swings in electricity prices. Li and Ierapetritou (2008) study te impact of tree different uncertainty sets proposed in te robust optimization literature for process sceduling under bounded parameter uncertainty: Soyster's worst case approac (1973), Ben-Tal and Nemirovski (1999) as well as Bertsimas and Sim (2003). Wile Soyster's formulation is considered to be too conservative, Ben-Tal and Nemirovski's (1999) approac introduces nonlinearities. Terefore, tey identify te polyedral uncertainty set of Bertsimas and Sim (2003) to be most promising because it provides a computationally tractable linear formulation, wic also allows controlling te degree of conservatism. Bertsimas and Sim's (2003) approac assumes tat eac uncertain parameter will be eiter at its nominal or worstcase value and tat te total number of parameters tat take teir worst-case values is restricted. In te context of uncertain electricity prices, Conejo et al. (2010) use Bertsimas and Sim s polyedral uncertainty set to optimize te demand response for electricity consumers using a rolling-orizon algoritm. Unfortunately, te polyedral uncertainty set as two disadvantages wen applied to electricity prices. First, te prices will eiter take teir nominal or teir worst-case values, wic is not te case for electricity prices in reality. Second, te number of parameters, wic will take teir respective worst-case value (also referred to as budget of uncertainty ), is ard to coose correctly. Te budget of uncertainty is a modeling parameter and describes te degree of conservatism. However, for electricity prices its appropriate value cannot be easily derived from istorical data. Recently, Duzgun and Tiele (2010) use multiple ranges for a more detailed modeling of te realizations of te uncertain parameters in te context of R&D project selection. Tey introduce bins wit different parameter ranges and assume tat eac parameter will fall in one bin, attaining its respective worst case witin te bin. Te number of parameters for eac bin is restricted and a modeling decision, e.g. observed from istorical data. In tis work, we utilize te uncertainty set by Duzgun and Tiele (2010) to build a computationally tractable uncertainty set for electricity prices based on istoric price data. We modify teir model to also account for correlations in te observed data. Te resulting uncertainty set is integrated in te operational planning model for air separation plants tat is described in Mitra et al. (2011) and a case study is solved to sow te impact. Problem Statement Given is a power-intensive process tat produces a set of products g G, wic can be partitioned into Storable and Nonstorable products. Te plant as different operating modes m M depending on te equipment tat is running and is subject to ourly-varying electricity prices e (our H). A price forecast is given for te orizon of a week. Furtermore, istorical data for eac individual our of te week is available. Te problem is to find a weekly production plan for te power-intensive process, suc tat operational costs due to power consumption are minimized wile satisfying various operational constraints of te plant. At te same time, te probabilistic data sould be used to control te financial risk associated wit te obtained weekly production profile. Process Representation We represent te plant wit a set of modes m M, eac one aving a distinct feasible space of operation. In every our, te plant operates in exactly one mode. Furtermore, te dynamics of te plant require additional logic constraints tat e.g. ensure minimum stay relationsips (minimum up-/downtime) or forbid certain transitions between modes. Additionally, mass balances for te Storable products ave to be enforced to account for inventory levels. Te power consumption PW in our is approximated by a linear correlation between te operational decision variables x m,g of te plant using correlation coefficients! m,g as sown in Eq. (1b). Note tat te operational decision variables can be continuous (e.g. production levels) as well as discrete (e.g. state of te plant in terms of modes). Te exact derivation of all equations can be found in Mitra et al. (2011). For te purpose of tis paper, we state te deterministic problem of finding te scedule wit minimum costs tat satisfies a pre-defined demand for a given forecast of ourly electricity prices e te following way: min PW, x m,g! PW (1a) e s.t. PW =!! m,g x m,g x m,g m,g (1b)! X (1c) Note tat te operational constraints of te problem are summarized in te set of constraints X as described in Eq. (1c). Robust Optimization Te general idea of robust optimization is to robustify te solution of an optimization problem wit respect to uncertainty in te problem data tat can be in cost coefficients, in te constraint matrix or te rigt-and side

3 (Bertsimas and Sim, 2004). In our case, we consider uncertainty in te cost coefficients e. We assume tat te uncertainty for e is bounded in te interval [E(e )! ê, E(e )+ ê ], were E(e ) is te nominal (expected) value for te electricity price at our and ê te expected maximal deviation from te nominal value. All e are part of te uncertainty set E, wose exact structure is a degree of freedom for te modeler. Te robust optimization problem of te deterministic problem in (1) wit uncertain cost coefficient can be written as: min max PW, x m,g " e PW (2a) e!e Uncertainty Set (2b) Te appropriate selection of te uncertainty set E is a non-trivial task due to concerns regarding tractability and conservatism. As stated earlier, we study te influence of te uncertainty set described by Duzgun and Tiele (2010) tat features multiple ranges for a more detailed modeling of te realizations of te uncertain parameters. It as te advantage tat it allows to tune te degree of conservatism according to istorical data wile providing a formulation tat is tractable. Te formulation is te following: min max e PW, x m,g e,w k! PW (3a) (3b) e k w k! e k! e k w k, " # H, k # K (3c) e = # e k,! " H (3d) # w k =1,! " H (3e) # w k! " k, $k % K (3f) w k! {0,1}, "! H, k! K (3g) It is assumed tat te uncertainty space can be split up into multiple ranges k! K. Te ranges represent relative deviations from te mean value for eac individual parameter e. Eac e falls in exactly one bin k (see Eq. (3c)-(3e)). Te number of parameters per bin is limited by te parameter! k as stated in Eq. (3f). Te binary variable w k describes weter e lower and upper bounds e k falls into bin k (Eq. (3g)). Te and e k are constructed wit respect to te relative deviations of te ourly prices from teir corresponding mean values. Based on te analysis of istorical data for a given season of te year, we generate te bins for te relative deviations of te ourly prices for a wole week and define te allowed observations per bin accordingly. In formulation (3), we do not account for correlations in te random variables across different ours. However, in reality tese correlations exist, i.e. if te electricity price is already ig in te morning of a summer day it is likely to increase trougout te day. Terefore, similarly to Bertsimas and Sim (2004), we introduce te uncertain variables c i, wit i! I being te set of independent random variables. Te electricity prices can be rewritten as e =! c i! i, wit te parameter! i describing te i influence of c i on te ourly price e. We formulate te robust optimization problem over te uncertainty set of te independent random variables C, wic is analogously described to E wit te following set of equations: min max c PW i! i, x m,g c i,z i,k!! PW (4a) i (4b) c i,k z i,k! c i,k! c i,k z i,k, "i # I, k # K (4c) c i = # c i,k,!i " I (4d) # z i,k =1,!i " I (4e) $ z i,k! " k, %k # K (4f) i#i z i,k! {0,1}, "i! I, k! K (4e) Similarly to w k, te binary variables z i,k indicate weter c i falls into bin k or not. In te LP relaxation of formulation (9)-(14) z i,k is relaxed as 0! z i,k!1, "i # I, k # K (4g) Robust Counterpart In order to derive te robust counterpart of formulation (4), te linear relaxation of te inner maximization is investigated. As sown by Duzgun and Tiele (2010), te integer variables z i,k will take integer variables in te solution of te linear relaxation. Te key idea of te proof is te assignment polytope structure tat can be found in Eqns. (4e)-(4f). Furtermore, z i,k will take its maximum value in its range since te power consumption PW is nonnegative. Terefore, it follows tat c i = " c i,k z i,k for all i and te robust counterpart can be k!k written as: min "! i + "" k! k + "# i,k (5a) i#i k i,k (5b)! i +" k +# i,k! # c i,k $ i PW, $i " I, k " K (5c) "H! k, " i,k! 0, "i # I, k # K (5d) In tis formulation (5),! i is te dual multiplier for Eq. (4e),! k is te dual multiplier for Eq. (4f) and! i,k stems from te upper bound constraint in te LP relaxation of Eq. (4g).

4 Case Study Setup Air separation plant Te air separation plant tat we consider as two liquefiers tat produce te storable products (liquid oxygen, nitrogen and argon) and does not ave a pipeline customer. If te second liquefier ramps up production, te plant is 4 ours in a transitional state in wic almost no product is produced. Furtermore, once te plant is sut down, it as to stay offline for at least 24 ours. After a start-up, wic takes 4 ours, te plant as to run for at least 48 ours. We use industrial data provided by Praxair to conduct te case study. We assume tat te plant s liquid product inventory is at 40% at te beginning of te week, and te same value is used for te target inventory at te end of te week. Every 6 ours a constant pre-defined amount of product is removed from te storage tank. Te demand is equivalent to 72% capacity utilization. original variables to te transformed variables. Eac row of V is scaled to unit lengt. All entries in U and V are nonnegative, wic is of particular importance wen te dual of te inner maximization in (4) is taken. Since te data matrix A represents te relative deviation from te nominal (forecasted) value E(e ), we can construct our coefficients! i as follows:! i = v i, E(e ) (7) For our case study, we reduce te 168 ours of a week to 10 independent random variables. Te distribution in te transformed variables is sown in Figure 2. Electricity Data In tis case study we analyze electricity data on an ourly basis for te PJM day-aead market (Ott, 2003). In Figure 1, we can see te distribution of te relative deviations from teir ourly seasonal average for te spring seasons Te grap was generated considering 13 weeks of te spring season for eac year. Figure 2. Distribution of deviations in te transformed variables Te frequencies for eac bin (scaled by te number of independent random variables, ere 10) are taken as te input for te! k parameters in our uncertainty set model. Results Figure 1. Distribution of relative deviations from nominal values ( 1 ) in electricity prices for a typical week in spring. To account for correlations in electricity prices, we ave to use a dimension-reduction tecnique. We coose Nonnegative Matrix Factorization (NMF, Berry et al., 2007), wic performs a low-rank approximation of te data matrix A in te following way: A! U V (6) Te columns of U represent transformations of te variables in A. V describes te relative contributions of te We solve te deterministic and te robust optimization problem, and can observe tat bot solutions beave similarly wit a few exceptions tat we will explain in te following. Figure 3 sows te nominal values for te electricity prices of one week as well as minimum and maximum values. Here te nominal values are te means for eac our of te spring of As we can see in Fig. 4, te power consumption is countercyclical to te electricity prices in bot solutions. During te nigt, wen electricity is ceap, te second liquefier is activated (above 0.7 relative power consumption). During te first 48 ours, tere are only sligt differences between te two solutions wit respect to te timing of te start-up of te second liquefier. For te following tree days, te main difference is te sutdown of te second liquefier in te robust solution to reduce te exposure to te market volatility on Wednesday morning (our 55). Over te weekend, we can observe tat te robust solution delays te start-up of te second liquefier on Friday nigt (our 117), and keeps it running during Saturday wile exploiting te lower volatility during ours

5 4 CONTROLLERS Figure 3. Electricity prices for a typical week in spring 2008 (mean, max and min values) Figure 4. Power consumption profiles for deterministic and robust solution From te profiles it is not immediately obvious wic solution sould be implemented. Hence, we compare te two solutions wit respect to 2000 electricity profiles tat were randomly generated based on te relative deviations, wic were observed in data. In Figure 5, we can see tat bot solutions perform similarly wit te robust solution aving sligtly smaller tails. Table 1. Cost comparison for randomly generated electricity profiles. Deterministic Robust Difference Mean $121,430 $121, % Standard deviation $10,098 $9, % Computational Results Te model statistics are sown in Table 2. It is interesting to note tat te problem size does not increase significantly. Wit I =10 (number of uncertain parameters) and K =10 (number of bins), we introduce only 100 new constraints and 120 new continuous variables. Te number of binary variables does not cange. We use CPLEX witin te GAMS modeling environment (Brooke et al., 2010) to solve te resulting MILP problems on a MacBook Pro wit a 2.53 Gz Intel Core i5 and 4GB RAM. Te CPU time required increases from 6 s (deterministic) to 16 s (robust). Table 2. Model statistics for case study. Figure 5. Comparison of cost distributions for randomly generated electricity profiles. In Table 1, we can observe tat te standard deviation decreases by 2%, wile te mean of te robust solution increases by 0.1% (i.e. te price of robustness ). 5 Deterministic Robust # Constraints 42,163 42,263 # Variables 29,401 29,521 # Binary 3,528 3,528 CPU time 6 s 16 s

6 Conclusion We ave applied robust optimization to te sceduling of power-intensive processes under uncertain electricity prices. Te uncertainty set featured multiple ranges and accounted for correlated data. For an illustrative case study wit an air separation plant, we observed tat te robust solution beaved similarly to te deterministic solution wit a few exceptions. Te differences caused te robust solution to ave a 2% lower standard deviation in te cost distribution tat was calculated for randomly generated electricity profiles. At te same, te mean of te robust solution increased by only 0.1%. Wile tese numbers suggest tat te robust solution migt be wort it to implement, te results can also be interpreted in a way tat te deterministic solution itself is a somewat robust solution. Hence, it would be interesting to study te performance of te deterministic and te robust solution witin a rolling orizon sceme as suggested by Conejo et al. (2010). Processes. Accepted to Computers & Cemical Engineering. Ott, A. L. (2003). Experience wit PJM Market Operation, System Design and Implementation. IEEE Transactions on Power Systems, 18, 528. Soyster, A. L. (1973). Convex programming wit set-inclusive constraints and applications to inexact linear programming. Operations Researc, 21, Acknowledgments We tank te Center for Advanced Process Decisionmaking (CAPD) at Carnegie Mellon University and Praxair for financial support. References Ben-Tal, A.; Nemirovski, A. (1999). Robust solutions to uncertain programs. Operations Researc Letters, 25, 1. Berry, M. W.; Browne, M.; Langville, A. N.; Pauca, V. P.; Plemmons, R. J. (2007). Algoritms and Applications for Approximate Nonnegative Matrix Factorization. Computational Statistics and Data Analysis, 52, 155. Bertsimas, D.; Sim, M. (2003). Robust Discrete optimization and Network Flows. Matematical Programming, 98, 49. Bertsimas, D.; Sim, M. (2004). Te price of robustness. Operations Researc, 52(1), 35. Brooke, A.; Kendrick, D.; Meeraus, A. (2010). GAMS: A users guide, release Sout San Francisco: Te Scientific Press. Conejo, A.J.; Morales, J.M.; Baringo, L. (2010). Real-Time Demand Response Model. IEEE Transactions on Smart Grid, 1, 236. Duzgun, R.; Tiele, A. (2010). Robust Optimization wit Multiple Ranges: Teory and Application to R & D Project Selection. Submitted to European Journal of Operational Researc, Available at ttp:// Ierapetritou, M.G.; Wu, D.; Vin, J.; Sweeny P.; Cigirinskiy M. (2002). Cost Minimization in an Energy-Intensive Plant Using Matematical Programming Approaces. Industrial & Engineering Cemistry Researc, 41, Li, Z.; Ierapetritou, M. G. (2008). Robust Optimization for Process Sceduling Under Uncertainty. Industrial & Engineering Cemistry Researc, 47, Mitra, S; Grossmann, I. E.; Pinto, J. M.; Arora, N. (2011) Optimal Production Planning under Time-sensitive Electricity Prices for Continuous Power-intensive