A MINLP Model for a Minimizing Fuel Consumption on Natural Gas Pipeline Networks

Transcription

1 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle A MINLP Model for a Mnmzng Fuel Consumpton on Natural Gas Ppelne Networks Dana Cobos-Zaleta Roger Z. Ríos-Mercado Graduate Program n Systems Engneerng Unversdad Autónoma de Nuevo León AP 111-F, Cd. Unverstara San Ncolás de los Garza, NL 66450, Méxco {dana,roger}@yalma.fme.uanl.mx Abstract: The problem of mnmzng fuel consumpton on natural gas ppelne networks s addressed. A mxed-nteger nonlnear programmng model for a specal case of ths problem wll be presented and dscussed. In addton, our computatonal experence on evaluatng an outer approxmaton wth equalty relaxaton and augmented penalty method s shown. The results, usng dfferent networks topologes over dfferent type of compressor unts, show how ths model can be solved effectvely. Key words: Mxed-nteger nonlnear programmng, natural gas, ppelne networks 1. Introducton Natural gas s transported by pressure throughout a ppelne system. Ths transmsson produces energy loss caused by the exstng frcton between the gas and the ppelne's nner wall, and for the heat transfer between the gas and the envronment. Compressor statons nstalled n the network compensate for ths energy loss by ncreasng the pressure to keep the gas movng. Typcally, the compressor statons consume n fuel about 3 to 5 % of the total gas flown through the network (Wu, 1998). Ths becomes sgnfcant as about thousand of mllons of cubt feet of gas are transported every day. Hence the mportance of fndng a better way to operate these compressor statons through a ppelne system. There are several varatons of ths problem dependng on the modelng assumptons and the decsons to be made. One of the modelng assumptons made n most of the prevous works s that the number of compressor unts to be workng wthn each compressor staton s fxed. In our work, we consder ths as a decson varable hence the model becomes a mxed-nteger nonlnear problem (MINLP). The problem s typcally modeled as a non-lnear network flow problem where decson varables are mass flow rate at each arc and pressure drop at each node. Examples of ths representaton are shown n Fgures 3, 4, and 5, where the arcs represent ether compressor statons or ppelnes and the nodes represent supply, transshpment or demand ponts. In ths work we present a MINLP model for the problem of mnmzng the fuel consumpton n a ppelne network. Our decson varables are the pressure at each node of our network, the mass flow trough the ppelne, and the number of compressor unts that have to be on wthn each staton. We present a computatonal experence by evaluatng an outer approxmaton wth equalty relaxaton and

2 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle augmented penalty method, whch solves two knds of problem: one called the master problem for solvng the non-lnear constrans and the sub-problem, whch consders the mxed-nteger part. Ths methodology can handle the fact that the objectve functon or the feasble doman can be non-convex. See Floudas (1995). In our prelmnary fndngs, we have seen that t s possble to solve small problems for certan knd of compressor unts optmally, specally when applyng a pre-processng phase (scalng the equatons), but t s qute complcated fndng a feasble soluton for the others. 2. Problem Descrpton These are the modelng assumptons. We assume that the problem s n steady state. Ths s, our model wll provde soluton for systems that have been operatng for a relatve large amount of tme. Transent analyss would requre ncreasng the number of varables and the complexty of ths problem. The network s balanced. Ths means that the sum of all the net flows n each node of the network s equal to zero. In other words, the total supply flow s drven completely to the total demand flow, wthout loss. We know that compressor statons are feed wth some of the fuel drven trough the ppelnes, and for sustanng ths assumpton we consder the cost of ths consumpton as an extra cost n our model named opportunty cost that represents the amount we should spend f we bought the fuel from thrd partes. Each arc n the network has a pre-specfed drecton. There are a pre-specfed number of dentcal centrfugal compressors connected n parallel n each compressor statons. 2.1 Model In ths work, parameters and data are represented wth upper case letters, whle varables are represented n lower case. Parameters: Vs: Set of supply nodes Vd: Set of demand nodes V: Set of all nodes n the network Ap: Set of ppelnes arcs Ac: Set of compressor staton arcs A: Set of all arcs n the network; A = Ap Ac U : Arc capacty of ppelne (,; (, Ap R : Resstance of ppelne (,; (, Ap N : Upper bound on the number of compressor unts staton (,; (, Ac L U P, P : Pressure lmts at each node; L = lower bound, U= upper bound; V b : Net mass flow rate at each node; b > 0 f Vs, b < 0 f Vd, b = 0 otherwse Varables: x : p : n : Mass flow rate n arc (,; (, A Pressure at node ; V Number of compressor unts workng at staton (,; (, Ac

3 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle Formulaton: Objectve functon Balance flow equaton n each node, where V b = 0 mn g (, x, p, p j ) (, Ac { j (, A} { j (, A} ( (1) x x = b j Ppelne capacty x U (, Ap Gas flow dynamcs n each ppelne (steady state) p p = R x j (, Ap Pressure range P L p P U V Operatonal lmts at each compressor staton x n, p, p j D (2) (, Ac { 0,1,2 N } x, p 0, n,..., (3) It s mportant to menton that a compressor staton s composed of several dentcal centrfugal compressors, connected n parallel that mght be turned on or turned off, see Fgure 1. Compressor 1 Compressor Staton Compressor 2... Compressor N Fgure 1. Representaton of a compressors staton For a sngle centrfugal compressor unt (,, ts doman s determned by the varables x (flow through the arc ), p (nlet pressure) and p j (outlet pressure). Now, when consderng N unts wthn the staton, the flow x through the staton can be equally splt nto the number of compressor statons workng. The flow trough each unt becomes x /n so

4 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle x, p, p n (1998). j must satsfy D from equaton (2). A more detaled descrpton can be found n Wu So t has been found (Wu et. al, 2000) that the doman D of a centrfugal compressor (, s defned by: h q q q 2 = A + + H + BH C H D H s s s s From prevous work (Wu et al., 2000) constrant (2) can be expressed as: 2 3 h m ZRT s p = 1 m p j x q = ZRTs p where the followngs parameters are assumed known wth certanty: A H, B H, C H, D H Constants, whch depend on the type of compressor (typcally estmated by least square method). T s Gas temperature Z Gas compressblty factor R Gas constant m = (k-1)/k, where k s the specfc rato R L Surge (lower lmt of q /s ) R U Stonewall (uper lmt of q /s ) and the followng auxlary varables are ntroduced: q Inlet volumetrc flow rate n compressor (,; (, є Ac h Adabatc head of compressor (,; (, є Ac s Compressor speed that should between S mn S S max, where speed S mnumun speed and S max = maxmum speed are known. Varables h, q and s are drectly known to the operator; however, gven the mappng from (h, q, s ) to (x, p, p j ), t s preferable to work on the latter space from the network optmzaton perspectve. Fgure 2 llustrates ths doman n the (x, p, p j ) space for x fxed. mn =

5 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle Fgure 2. Doman of a compressor unt, wth x fxed n 6000 lbm/mn As we can apprecate of the doman for a centrfugal compressor s non-convexty. It s know from prevous work (Vllalobos-Morales and Ríos-Mercado, 2002) that a good approxmaton to the real cost functon s gven by: 2 2 x p j x p j x p j g ( x, p, p j ) = x A6 B6 C6 D6 E6 + F6 p p p p p p where A 6,, F 6 are known constants. It s well know that the behavor of each compressor s non-lnear. Furthermore, the feasble doman n (2) s a non-convex set. In addton, the objectve functon s also non-convex. These features make ths problem partcularly nasty. Now, some MINLP solvers wll allow bnary varables only. In that case, the model would have to be modfed n the followng way. A bnary varable n k, whch s equal to one f the k-th compressor of staton compressor (, s workng, and 0 otherwse. Then we add the equaton x = n (, Ac ; and allow n to become a real varable. k k, 3. Prevous Work 3.1 Fxed Number of Compressor Unts We now hghlght the most relevant contrbutons addressng the specal case where the number of unts s fxed and therefore not a varable n the model. From the optmzaton perspectve, most of the approaches have been based on dynamc programmng technques. The man advantages of DP are that a global optmum s guaranteed and that no lnearty can be easly handled. Dsadvantages of DP are that ts applcaton s practcally lmted to networks wth smple

6 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle structures, such as lnear or tree-lke topologes (see Fgures 3 and 4), and that computaton ncreases exponentally n the dmenson of the problem, usually refered as the curse of dmensonalty. In topologes wth no cycles, t has been showed that the flow varables can be unquely determned and thus elmnated from the problem. DP then focuses on fndng an optmal set of pressures. Among the most relevant work we can cte Wong and Larson (1968), Lall and Percell (1990), and Carter (1998), who worked on a nonsequental DP algorthm to handle cyclc networks when the mass flow rate varables are fxed. For a more detaled descrpton of DP appled to gas networks, the reader s refered to Ríos-Mercado (2002). Gradent search technques, such as the generalzed reduced gradent method are also a choce. Advantages of the GRG method are that t avods the dmensonalty ssue and that t could be appled to networks wth cycles. However, snce the GRG method s based on a gradent search method, there s no guarantee to fnd a global optmum, especally n the presence of dscrete decson varables, so t may stall at local mnma. The most sgnfcant work n ths respect s due to Percell and Ryan (1987). Other related work nclude Osadacz (1987), who worked on numercal smulatons of gas ppelne networks wth no optmzaton nvolved; Osadacz and Swerczewsk (1994) and Osadacz (1995), who used herarchcal optmzaton technques; Wu, Boyd and Scott (1996), who used a mathematcal model for the fuel cost mnmzaton over a sngle unt compressor staton; Km, Ríos-Mercado, and Boyd (2000), who proposed an approxmaton algorthm that teratvely adjusts the flow varables n a heurstc way and then fnds an optmal set of pressures; and Ríos-Mercado et al. (2002), who develop a technque to reduce the sze of the network at pre-processng. 3.2 Number of Unts Not Fxed To the best of our knowledge, the only work dealng wth the number of unts as a varable s that of Wu et al. (2000). However ther model s not qute a MINLP. They frst determnate, at frst level, the amount of flow through the compressor staton, and then, at a second level, fgure out the optmal number of unts for that partcular flow. That approach of course lmts the search for a global optmum. Snce our dea s treat all varables, at the same level, ths s what motvates the choce of handlng ths problem as a MINLP, whch becomes the man purpose of ths work. 4. Proposed Soluton Procedure As we have seen n the prevous secton, some researchers have consdered as decson varables the pressure drop at each node of the network and the mass flow transported n the ppelne. The varaton we are tryng to handle s to consder smultaneously that n each compressor staton there s a number of compressors connected n parallel and n dependence of the flow, we wll decde how many compressor to turn on for transportng the fuel. Ths means addng another decson varable of the nteger type. We wll try to solve ths knd of problem consderng smultaneously both varables types (contnuous and nteger), whch makes ths problem a MINLP. Among the most popular methodologes for solvng MINLP models we fnd: 1. Generalzed Benders Decomposton (GBD) 2. Branch and Bound (BB) 3. Outer Approxmaton (OA) 4. Feasblty Approach (FA) 5. Outer Approxmaton wth Equalty Relaxaton (OA/ER) 6. Outer Approxmaton wth Equalty Relaxaton and Augmented Penalty (OA/ER/AP)

7 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle These approaches are better descrbed n Floudas (1995). We have chosen the outer approxmaton wth equalty relaxaton and penalty augmented method because ths can handle the non-convexty n the objectve functon, the doman or both. We know however, gven the non-convexty of our model, global optmums are not necessarly guaranteed (Floudas, 1995). The OA/ER/AP method, due to Grossmann et. al (2001) at the Engneerng Desgn Research Center (EDRC) at Carnege Mellon Unversty, s mplemented n a software called DICOPT. DICOPT (Grossman et al., 2001) s a solver avalable n GAMS (Brooke, Kendrck, and Meeraus, 1992) for solvng MINLPs. The algorthm solves teratvely a seres of NLP and MIP sub-problems. In the full verson of the paper we wll nclude a detaled dscusson of the algorthm, and hghlght the algorthmc parameters that were evaluated. 5. Computatonal Work The purpose of ths work s twofold. Frst, we would lke to be able to solve a large number of nstances of ths problem and to show better solutons can be reached than those obtaned by approaches that consder a fxed number of unts wthn each compressor. Then, we evaluate the performance of algorthmc parameters to asses the effectveness of the method on ths type of problems. Ths wll nclude fndng the best parameters that yeld hgh qualty solutons. In order to do that, we have mplemented the model n GAMS. Frst, we consder a smple topology (see Fgure 3), whch conssts of 6 nodes (one demand, one supply), 5 arcs (2 compressors and 3 ppelnes). For ths topology, 9 dfferent types of compressors, wth data taken from real-world unts, were tested. The model was run on a Sun Ultra 10 under Solars 7 OS. 1 2 E 3 4 E 5 6 Fgure 3. Lnear topology. We frst ran the problem settng net flow values of 400 MMCFD (1 MMCFD = 10 6 cubc feet per day) and found numercal dffcultes. Only two of nne compressors were solved. In other nstances, we found that some Jacobean elements were too large, so that the algorthm was unable to fnd a soluton. So we ncreased the flow to 950 MMCFD and appled a pre-processng phase, whch conssted of scalng some of the constrants. The results are shown n Table 1. As can be seen the algorthm found optmal or feasble solutons n 5 of the 9 nstances. Ths llustrates the mportance of an approprate scalng n the preprocessng phase, but t also shows further work s necessary at pre-processng to derve models wth no numercal dffcultes. For the problems solved, we can also observed that most of the tme was spent on solvng the MINLP sub-problem. The compressor s name s allocated n the frst column n Table 1. The model status column ndcates the stoppng crtera used by DICOPT, where Intermedate non nteger means that the solver faled n the NLP sub-problem, Integer soluton means that the solver was able to fnd feasble soluton, and Locally optmal means that a local optmal soluton was found. The thrd column shows the numercal value of the objectve functon that represents the consumpton cost. The ffth column

8 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle shows how long the solver takes to fnd the soluton, and the lasts two columns show the total tme and percentage taken by each sub-problem. Compressor type Cdbnk1 Cdbnk3 Cdbyk2 Cdryk1 Cdsnk1 Cdbnk2 Cdbyk1 Cdhrk1 Cdryk2 Model status Intermedate non nteger Integer soluton Intermedate non nteger Intermedate non nteger Locally optmal Intermedate non nteger Integer soluton Integer Soluton Integer soluton Objectve functon Number of teraton Duraton CPLEX (tme, %) CONOPT (tme, %) Table 1. Results of expermentaton In those nstances where the algorthm faled to fnd a soluton, t has been observed that the maxmum teraton number s reached, and the soluton s nfeasble. That happens when an NLP sub-problem cannot be solved to optmalty. Some NLP solvers termnate wth a status other than optmal f not all of the termnaton crtera are met. For nstance, the change n the objectve functon s neglgble (ndcatng convergence) but the reduced gradents are not wthn the requred tolerance. Such a soluton may or may not be close to the (local) optmum. Another explanaton s that the NLP sub-problem fals resultng n a non-optmal but feasble solutons. Sometmes an NLP solver cannot make further progress towards meetng all optmalty condtons, although the current soluton s feasble. Further work s under way now to attempt to explot the current problem structure so we can deal wth these dffcultes successfully. Ths s an ongong research. We are stll workng on pre-processng to address the numercal dffcultes obtaned when applyng the algorthm. It s expected that the full verson at the paper wll contan results for all nstances. In addton, the full verson of the paper wll contan optmal results (not shown here) for other type of topologes (llustrated n Fgures 4 and 5) and a comparson to the approach, whch uses a fxed number of compressors. Acknowledgments: Ths research s supported by the Mexcan Natonal Councl for Scence and Technology (CONACYT grant J33187-A) and Unversdad Autónoma de Nuevo León through ts Scentfc and Technologcal Research Support Program (PAICyT grants CA and CA763-02).

9 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle Fgure 4. Example of a tree topology Fgure 5. Example of a topology wth cycles. References A. BROOKE, D. KENDRICK, AND A. MEERAUS (1992). GAMS: A User s Gude, Release Scentfc Press, San Francsco. R. G. CARTER (1998). Ppelne optmzaton: Dynamc programmng after 30 years. In Proceedngs of the 30th PSIG Annual Meetng, Denver,October. C. A. FLOUDAS (1995). Nonlnear and Mxed-Integer Optmzaton Fundaments and Applcatons. Oxford Unversty Press, New York. I. E. GROSSMAN, J. VISWANANTHAN, A. VECCHIETTI, R. RAMAN, AND E. KALVELAGEN (2001). GAMS/DICOPT: A Dscrete Contnuous Optmzaton Package. H. S. LALL AND P. B. PERCELL (1990). A dynamc programmng based gas ppelne optmzer. In A. Bensoussan and J. L. Lons (edtors), Analyss and Optmzaton Systems, Volume 144, Lecture Notes n Control and Informaton Scences, pp , Sprnger-Verlag, Berln. A. J. OSIADACZ (1987). Smulaton and Analyss of Gas Netwoks, Gulf Publshng, Houston.

10 Memoras del XI Congreso Latno Iberoamercano de Investgacón de Operacones (CLAIO) de Octubre de 2002 Concepcón, Chle A. J. OSIADACZ (1995). Dynamc optmzaton of hgh pressure gas networks usng herarchcal systems theory. In Proceedngs of the 26th PSIG Annual Meetng, Albuquerque, October. A. J. OSIADACZ AND S. SWIERCZEwsk (1994). Optmal control of gas transportaton systems. In Proceedngs of the 3rd IEEE Conference on control Applcatons, pp , August. P. B. PERCELL AND M. J. RYAN (1987). Steady-state optmzaton of gas ppelne network operaton, In Proceedngs of the 19th PSIG Annual Meetng, Tulsa, OK, October. R. Z. RÍOS-MERCADO (2002). Natural gas ppelne optmzaton. In P. M. Pardalos and M. G. C. Resende, edtors, Handbook of Appled Optmzaton, Chapter , Oxford Unversty Press, New York. R. Z. RÍOS-MERCADO, S. WU, L. R. SCOTT Y E. A. BOYD (2002). A reducton technque for natural gas transmsson network optmzaton problems. Annals of Operatons Research. Forthcomng. Y. VILLALOBOS-MORALES AND R. Z. RÍOS-MERCADO (2002). Approxmatng the fuel consumpton functon on natural gas centrfugal compressors. In Proceedngs of the NSF Desgn, Servce, Manufacturng and Industral Innovaton Research Conference, San Juan, Puerto Rco. P.J. WONG AND R.E. LARSON (1968). Optmzaton of natural-gas ppelne systems va dynamc programmng. IEEE Transactons on Automatc Control, AC-13 (5): S. WU (1998). Steady-State Smulaton and Fuel Cost Mnmzaton of Gas Ppelne Networks, Ph.D. dssertaton, Unversty of Houston, Houston, August. S. WU, R. Z. RÍOS-MERCADO, E. A. BOYD, AND L. R. SCOTT (2000). Model relaxatons for the fuel cost mnmzaton of steady-state gas ppelne networks. Mathematcal and Computer Modelng, 31(2-3):