Pareto Optimization with Reverse Normal- Boundary Intersection for Power Plant Models

Transcription

1 1 Pareto Optmzaton wth Reverse Normal- Boundary Intersecton for Power Plant Models Joel H. Van Sckel and Kwang Y. Lee, Fellow, IEEE Abstract Ths paper presents the use of Pareto optmzaton technques that was prevously analyzed wth small scale power plant models where results could be verfed analytcally. Further research has been conducted nto the use of applyng these technques to large scale models and analyzng performance. These approaches have been verfed to scale well to larger applcatons. Through the use of mult-objectve partcle swarm optmzaton and reverse normal-boundary ntersecton, reference governors can be developed for numerous types of power plant. Combnng these two approaches allows decsons made from a Pareto front to be seamlessly mapped nto a smple and fast sngle varable optmzaton process. These applcatons are demonstrated wth a small ol-fred power plant model and a large-scale coal-fred power plant model. Index Terms Mult-objectve optmzaton, reverse normalboundary ntersecton, steady state power plant operaton. M I. INTRODUCTION ULTI-OBJECTIVE optmzaton s mplemented dfferently from area to area, wth numerous approaches avalable. Generally, most of these approaches can be splt nto two broad categores. The frst category, known as aggregaton technques, aggregates the multple objectves quanttatvely nto a sngle objectve. The second category, Pareto technques, uses the concept of Pareto optmalty to fnd a trade-off curve among multple objectves. Aggregaton of multple objectves can be acheved through dfferent rankng schemes such as weghted summaton [1] or goal programmng [2] to create a sngle varable optmzaton problem. Ths allows standard optmzaton technques to be utlzed. Unfortunately, quanttatvely aggregatng objectves assumes a quanttatve knowledge of the relatve mportance of the dfferent objectves. Addtonally, these technques only return a sngle pont as a soluton, whch can be a postve or a negatve dependng on the applcaton. For Pareto technques, fndng the trade-off curve, n whch numerous ponts are returned, provdes a comprehensve vew of the optmzaton problem, but the large number of ponts s hard to vsualze when more than three objectves are used. Furthermore, most applcatons need a specfc soluton, not a Ths work was supported by the NDSEG fellowshp program through the Offce of Naval Research and ASEE, and NSF Grant ECCS J. H. Van Sckel s wth the Department of Electrcal Engneerng, the Pennsylvana State Unversty, Unversty Park, PA (emal: jhv107@psu.edu). K. Y. Lee s wth the Department of Electrcal and Computer Engneerng, Baylor Unversty, Waco, TX (e-mal: Kwang_Y_Lee@baylor.edu). set of ponts, to functon properly. Ths creates a need to know how to choose a pont on the trade-off curve. Currently, mult-objectve optmzaton work for power plants has been acheved through the desgn of reference governors [3, 4]. Reference governors have used aggregaton technques to create a sngle objectve functon from multple objectves n order to control power plants. Unfortunately, there s no explct way to weght the dfferent objectves, and wthout lookng at the Pareto front, t s unknown what tradeoffs are beng made whle choosng the weghtng scheme. The Pareto front of a power plant was analyzed n [5]. Whle the approaches showed accurate ways to generate the Pareto front of a power plant, t was not enough of a step to provde somethng that could be mplemented for onlne operaton. A Pareto front by tself manly serves as a method to allow humans to better vsualze a mult-objectve optmzaton problem. How a specfc pont s chosen for operaton s often subjectve, f not even arbtrary. For ths applcaton of power plant control and specfc objectves, choosng the pont turned out to be qute smple. However, there was stll the ssue of how to use ths pont to control power plant operaton under changng operatng condtons. Smply runnng the power plant close to orgnal operatng condtons s not acceptable as the operatng front changes dramatcally as the unt load demand (uld) changes, completely changng where the optmzaton pont would be located. To address ths ssue n a manner that allows a pont to be chosen only once and not for every possble value of uld, a modfed verson of normal-boundary ntersecton was used. The methods dscussed n ths paper were appled to a smple thrd order power plant model as well as a hgher fdelty twenty thrd order model. Both are fossl fuel power plants. II. MULTI-OBJECTIVE OPTIMIZATION Mult-objectve optmzaton has always been mportant as t s not very often that there s an deal soluton that provdes the best result for every aspect of a problem. As such, great effort has gone nto tryng to develop methods to provde better ways to make decsons where tradeoffs are nevtable. When the problem can be quantfed, equatons can be used to gve numercal values to dfferent objectves representng how well they have been acheved. Intally, standard sngle objectve optmzaton technques were used, but they requred that dfferent objectves be aggregated. Ths was only a smplfcaton. To truly deal wth mult-objectve solutons n /10/$ IEEE

2 2 ther entrety, the concept of Pareto optmalty was put forth. A. Aggregaton Aggregaton technques n ther smplest form can be posed as the followng problem: mnmze ( x) [ ( x), ( x),..., ( x )] F = f f f 1 2 N Ths mult-objectve optmzaton problem can then be mapped nto a sngle varable optmzaton process: mnmze H( x) f ( x) 2 f ( x)... f ( x ) = α1 + α + + α 1 2 N N Where α n > 0 and H(x) s the scalar mappng of F(x). Ths s the most basc approach to aggregaton. B. Pareto Optmalty Pareto optmalty deals wth mult-objectve optmzaton where t s desred to mnmze the N objectves f (x), where = 1, 2,, N, x s a vector of length k, and k s the number of varables. Ths optmzaton can be posed as: mnmze ( x) [ ( x), ( x),..., ( x )] Subject to constrants: g ( x ) 0 h ( x ) = 0 F = f f f 1 2 N Termnology: Vector F(a) s sad to domnate vector F(b), F(a) p F(b), f and only f f (a) f (b) for all and f (a) < f (b) for at least one. Vector F(a) s consdered Pareto optmal or non-domnated f there exsts no other vector F(b) such that F(b) p F(a). There can be numerous ponts n a set that are all Pareto optmal. The collecton of all of the Pareto optmal ponts s called the Pareto front. An n-depth dscusson of Pareto optmalty s provded by [6]. III. POWER PLANT MODELS Two separate models are used n ths work. The frst model s a thrd order power plant often used for ts smple form and deal sze for publcaton. The second model s a twenty thrd order model that provdes a more challengng problem to ensure that approaches provded here can scale up to realworld applcaton. The thrd order model s presented fully n [7]. Ths model represents a 160 MW ol-fred turbne boler unt. The large scale power plant model was ntally developed by Usoro [8]. Ths model represents a 600 MW coal-fred drum type boler power plant. IV. APPLYING MULTI-OBJECTIVE OPTIMIZATION TO A POWER PLANT THROUGH A REFERENCE GOVERNOR Prevous work n ths area has mplemented mult-objectve optmzaton for power plant control though the use of a reference governor [3, 4]. A reference governor provdes feedforward control actons and set ponts to a feedback control system as shown n Fg. 1. Typcally, a heurstc optmzaton s used n conjuncton wth a steady-state neural network model to search for deal feedforward controls and setponts. Currently, multple objectves are aggregated by optmzng a cost functon that uses ether weghted sums [3] or goal programmng [4]. ULD...u n Output Reference Power Plant Governor Setponts Fg. 1. Tradtonal Reference Governor for power plant. Feedback Controller V. PARETO FRONT OF A POWER PLANT It s mportant that an accurate Pareto front of the power plant s operatng range be found. There are numerous approaches to ths process whch are beyond the scope of ths paper. To fnd ths front, the mult-objectve optmzaton problem must be stated n a quanttatve manner. For ths power plant, the mult-object optmzaton problem s stated below. x = [ uld, u, u, u ] (1) f ( x ) = uld P 1 (2a) f ( x ) = u (2b) 2 1 f ( x ) = u (2c) 3 2 mnmze F( x) = [ f ( x), f ( x), f ( x )] (3) Ths problem statement s suffcent for both models of power plants beng analyzed. For ths optmzaton problem f 1 s the error between unt load demand (uld) and power output (P) and f 2 represents fuel use, whch s controlled by the fuel valve openng. The fnal objectve functon f 3 represents the lfe of a control devce for u 2, whch s desred to be maxmzed for both types of power plants. Both power plants operate more effcently for hgher values of u 2. For the thrd order power plant, u 2 s the steam valve control whch s desred to be fully open, and for the twenty thrd power plant t s the steam flow rate, whch s also desred to be maxmzed. The results of ths optmzaton for the thrd order power plant can actually be determned analytcally [5], but n general a stochastc approach s requred to generate the Pareto front of these power plants [9]. There are numerous methods for generatng a Pareto front n exstence that work. After havng analyzed a few dfferent approaches [5], mult-objectve partcle swarm optmzaton (MOPSO) was chosen as a useful way to estmate the power plants Pareto fronts [9]. A. Mult-objectve Partcle Swarm Optmzaton MOPSO s based off of partcle swarm optmzaton (PSO)

3 3 whch one of the most wdely spread heurstc optmzaton technques based on ts smplcty and quck convergence. It has contnued to attract nterest n the area of mult-objectve optmzaton. There are many dfferent ways of modfyng PSO to work wth multple objectves. For ths work, the method by Alvarez-Bentez and others [10] s used. PSO uses a number of partcles that start wth random veloctes. These veloctes determne how the partcles move through the search space, and are updated every generaton usng v = w v + c r( Pbest x ) + c r ( Gbest x ) (4) c p p p For (4), v s the velocty vector, w, c 1, and c 2, are weghtng constants, and r 1, and r 2 are random varables between 0 and 1. For ths work, all constants are set to 0.5. The challenge n usng PSO for mult-objectve problems s to determne the values for the vectors Gbest and Pbest. Gbest s typcally the best pont currently found from all of the partcles and Pbest the best pont found for a specfc partcle. Snce multple ponts can be Pareto optmal, t s not mmedately apparent whch of the non-domnated ponts should be used for Gbest or Pbest because there s no best pont n a Pareto optmal set. Ths partcular mplementaton of MOPSO works by keepng a lst A, of all the non-domnated solutons found. After every generaton, the current partcles are combned wth A and then sorted by ther Pareto rank to create a new set of non-domnated ponts. A s then replaced wth ths new set. Pbest s ntalzed as the partcle s current poston, as n standard PSO, and updates to the partcle s new poston only f the new pont domnates Pbest. Gbest s dealt wth a lttle dfferently. Instead of havng a sngle Gbest, every partcle chooses randomly from A whch non-domnated pont t wll use for Gbest, and ths choce s made n every generaton. The results of MOPSO are shown n Fgs. 2-5 for both types of power plants. Ths algorthm generates far more ponts on the Pareto front than normal-boundary ntersecton. Ths slows the algorthm down as the number of ponts on the Pareto front grow, ncreasng the number of comparsons requred by the algorthm. The results are slghtly more nosy than that of NBI wth dfferental evolutonary (DE) algorthm but stll accurate. These results were acheved wth only 30 partcles over 100 generatons. B. Choosng a Pont Havng generated the Pareto front of a power plant, the next step s to choose whch pont s desred. For the thrd order power plant, the Pareto front s shown n Fgs. 2 and 3. For a gven error f 1 and uld each pont from Fg. 2 matches up wth the same pont n Fg. 3. As can be seen n Fg. 3, the desred locaton of changes dramatcally wth uld. For ths partcular power plant choosng ponts s easy. Except for extremely low power demands at whch ths power plant probably would not be runnng, we can always have u 2 set to 1 (fully open). As for a tradeoff, the power plant wll not purposely provde less power than needed to conserve fuel, so error wll be chosen to be 0 at all tmes, and then whatever correspondng to ths specfc Pareto pont wll be chosen. Smlar results are obtaned for the twenty-thrd order power plant and shown n Fgs. 4 and Fg. 2. Pareto front of thrd order power plant wth vs.. u Fg. 3. Pareto front of thrd order power plant wth u 2 vs.. (lb\s) Fg. 4. Pareto front of twenty thrd order power plant wth vs.. 30 MW 50 MW 75 MW 100 MW 130 MW 160 MW 30 MW 50 MW 75 MW 100 MW 130 MW 160 MW MW 500 MW 540 MW 580 MW

4 4 u 2 (lb\s) Fg. 5. Pareto front of twenty thrd order power plant wth u 2 vs.. VI. NORMAL BOUNDARY INTERSECTION Normal-boundary ntersecton (NBI) was developed by Das and Denns [11] for solvng mult-crtera optmzaton problems. It s geometrcally nspred and has been very successful n practce. Ths approach requres the knowledge of the soluton to the optmzaton of the ndvdual f (x), whch s usually avalable. In ths paper, NBI has been combned wth dfferental evoluton (DE) [12] to solve the optmzaton problem. Fundamentally, NBI s based on the concept of the Convex Hull of Indvdual Mnma whch s defned after a few prelmnares. For an n-objectve mnmzaton problem, where each objectve s represented by f (x), = 1,, n, let F = [f 1, f 2 f n ], where f s the soluton to the mnmzaton of f (x) alone, and x s the correspondng vector that mnmzes f (x). Let Φ be the matrx whose th column s F(x ) - F. The Convex Hull of Indvdual Mnma (CHIM) s then defned as the set of ponts that are convex combnatons of f. It may help to see t expressed n terms of (4): :, 1, 0 (4) NBI then proceeds by solvng the followng maxmzaton problem (5) by projectng lnes from the CHIM toward F whle varyng ω to search across the CHIM. max t xt, s.. t Φ ω + tnˆ = F( x) F h( x) = 0 g( x) MW 500 MW 540 MW 580 MW where ˆn s the unt normal vector to the CHIM and t represents how far ˆn can project from Φω before crossng the Pareto front. Typcally, all functons are shfted so that F les at the orgn. There are a few ssues unque to NBI that are mportant to be aware of. The above optmzaton only (5) searches over the CHIM smplex and could possbly leave out some extreme optmal ponts. In addton, a pont solved for usng NBI s not guaranteed to be Pareto optmal, however, every Pareto optmal pont can be represented as an NBI sub problem f ω s not restrcted to have ts elements sum up to 1. Regardless, NBI has proven to be a useful method of generatng Pareto ponts, and these ponts of possble concern are not an ssue n regard to reverse NBI as wll be explaned later. VII. REVERSE NORMAL-BOUNDARY INTERSECTION Whle NBI started wth a pont on the CHIM and then projected ˆn toward F, reverse NBI (RNBI) does the exact opposte and starts wth a pont that s known to be Pareto optmal, and projects ˆn toward the CHIM and uses the ntersecton to solve for a vector ω 0 that can be used to represent the Pareto optmal pont n terms of an NBI subproblem. For ths applcaton, ths represents the operatng ponts n a manner that s ndependent to the value of uld and allows for only a sngle pont to be chosen. Then, to recalculate ths pont under dfferent operatng condtons, ω 0 s used to solve a sngle NBI subproblem. Frst, a pont P 0 from the Pareto front must be chosen to represent the desred pont. Ths also means that the Pareto front needs to be generated. Ths can be seen as the frst step n RNBI as shown n Fg. 6. Everythng followng seeks to descrbe ths pont n terms of ts locaton n regard to the CHIM and as summarzed n the fgure. Bascally, ω o needs to be solved for as t s ndependent of the specfc values taken on by the Pareto front. To do ths, t s requred to fnd where ˆn would ntersect wth the CHIM when projected from P 0. Ths s done through a Gram-Schmdt orthonormalzaton and then projectng P 0 onto the CHIM smplex. To start wth, the n ponts f must be found as n NBI and defne the n-1 dmensonal affne subspace on whch the CHIM s located. Ths subspace s shfted to a lnear subspace by subtractng f 1 from the other n-1 ponts, puttng f 1 at the orgn and leavng n-1 vectors y, = 1,, n-1, wth n elements. These vectors can be turned nto an orthonormal bass usng Gram-Schmdt orthonormalzaton gven n (6-7): 1 k, = v y k k = 1 vk, vk v y v vˆ = v v Once there s an orthonormal bass, t s smple to fnd the ntersecton pont P. Frst P 0 should also be shfted by subtractng f 1 whch yelds a new pont P 0L, whch can be used wth the orthonormal bass. Usng (8) solves for pont P L, whch s easly transformed by to P by addng f 1. n 1, ˆ L = ol ol = 1 (6) (7) P P P v (8) Once P s known, t can be used to solve for ω 0 whch s then used to represent the desred Pareto optmal pont even f

5 5 the Pareto front s shfted. Ths can be used n all subsequent NBI subproblems to drectly fnd the equvalent P 0 even f the objectve functon space has been shfted, as n ths applcaton when the unt load demand changes. approxmately u 2 u Φω 0.4 tnˆ 0.2 Fg. 6. Second Step n Reverse NBI, solvng for ω tme (mn) Fg. 7. Feedforward controls for thrd order power plant. Now that ω 0 s known, the NBI maxmzaton problem n (5) can be solved substtutng ω 0 for ω. Choosng ω 0 n ths manner elmnates the prevous ssues that were mentoned as possble areas of concern when usng NBI to generate the Pareto front. As can be seen n (6), the NBI maxmzaton problem has nothng to do wth whether the optmal pont s domnated or not. It s only solvng for the pont that les n F whch can be projected by ˆn the furthest from the CHIM n the drecton of the orgn. For a non-convex surface, t s possble to have solutons to ths maxmzaton problem that are defntely not Pareto-optmal. VIII. IMPLEMENTING RNBI IN A REFERENCE GOVERNOR Implementng a reference governor wth RNBI s a farly straghtforward process [9]. Wth the mult-objectve problem stated, the Pareto front found, we must frst choose P 0. As stated before, we want to choose a pont wth an error of 0 and take whatever mnmum value of can be obtaned, and we know u 2 wll be maxmzed to 1 regardless of what pont we choose for the thrd order power plant [9]. We need to pck a specfc value for uld and pck a specfc P 0 so that ω 0 can be solved for. The unt load demand does not really matter as the results are gong to be almost dentcal regardless of ts value. All results are shown havng chosen P 0 when uld was set to 120 MW for the thrd order power plant, and 500 MW for the twenty thrd order power plant. From ths pont, RNBI can be followed exactly to yeld ω 0 and ths s then used to solve the NBI subproblem contnuously as uld vares. IX. NUMERICAL RESULTS An RBNI reference governor was developed for both power plants usng the prevously descrbed procedure. Pont P 0 was chosen wth uld = 120 MW for the thrd order power plant and chosen and uld = 500 MW for the twenty thrd order power plant. In both cases P 0 was chosen where error was Power (MW) Pressure (kg/cm3) tme (mn) tme (mn) Fg. 8. Thrd order power plant smulaton performance wth feedback control. (lb/s) Fg.9. Feedforward controls for twenty thrd order power plant. E Uld P Pd 62 tme (mn)

6 u 2 (lb/s) approach ntally developed scaled well when used on a sgnfcantly more complcated model that provded a much more data ntensve problem. Smlar results usng the same procedure were obtaned for both power plants even though the scale of number of nputs, outputs, and model complexty are vastly dfferent. 6 Fg.10. Feedforward controls for twenty thrd order power plant. Power (MW) Pressure (psa) Superheater Temp (R) tme (mn) tme (mn) Fg. 11. Twenty thrd order power plant smulaton performance wth feedback control. Fgs. 7-8 show the results for the thrd order power plant whch are reproductons from [9], and smlar performance was acheved for the large scale power plant n Fgs Only the most pertnent feedforward control actons are shown for the large scale power plant, but 12 control sgnals are actually generated. X. CONCLUSIONS The use of Pareto optmzaton technques that was prevously analyzed wth small-scale power plant models has been extended to large-scale power plant models and ther performances are analyzed. It was determned that the E uld P Pd T Td XI. REFERENCES [1] S. S. Rao, Engneerng Optmzaton Theory and Practce, 3 rd ed., [2] D. F. Jones and M. Tamz, Goal Programmng n the Perod , Multple Crtera Optmzaton: State of the Art Annotated Bblographc Surveys, 2002, pp [3] J. S. Heo, K. Y. Lee, and R. Garduno-Ramrez, Multobjectve control of power plants usng partcle swarm optmzaton technques, IEEE Transactons on Energy Converson, 2006, pp [4] R. Garduno-Ramrez and K. Y. Lee, Multobjectve optmal power plant operaton through coordnate control wth pressure set pont schedulng, IEEE Trans. on Energy Converson, 2001, pp [5] J. H. Van Sckel, P. Venkatesh, and K. Y. Lee, Analyss of the Pareto Front of a mult-objectve optmzaton problem for a fossl fuel power plant, IEEE PES General Meetng, [6] I. Y. Km and O. L. de Weck, Adaptve weghted sum method for multobjectve optmzaton: A new method for Pareto Front generaton, Structural and Multdscplnary Optmzaton, 2006, pp [7] R.D. Bell and K.J. Astrom. Smplfed Models of Boler-Turbne Unts. Report TFRT-3191, Lund Insttute of Technology, Sweden, [8] P. B. Usoro. Modelng and Smulaton of a Dum-turbne Power Plant Under Emergency State Control. M.S. Thess, Department of Mechancal Engneerng, MIT, Cambrdge, MA, [9] J. H. Van Sckel and K. Y. Lee, Reverse normal-boundary ntersecton for mult-objectve optmzaton for power plant operaton, Proc. of the IFAC Symposum on Power Plants and Power Systems Control, Tampere, Fnland, July 5-8, [10] J. E. Alvarez-Bentez, R. M. Everson, and J. E. Feldsend, A MOPSO algorthm based exclusvely on Pareto domnance concepts, EMO, 2005, pp [11] I. Das and J. E. Denns, Normal-boundary ntersecton: An new method for generatng the Pareto surface n nonlnear multcrtera optmzaton problems, SIAM Journal. on Optmzaton, 1996, pp [12] J. H. Van Sckel, K. Y. Lee, and J. H. Heo, Dfferental Evoluton and ts Applcatons to Power Plant Control, Proc. of Internatonal Conference on Intellgent Systems Applcaton to Power Systems, Kaohsng, Tawan, November 5-8, 2007, pp Joel H. Van Sckel receved hs B.S. degree n Electrcal Engneerng from Grove Cty College n 2005 and hs Ph.D. n Electrcal engneerng at the Pennsylvana State Unversty n Hs nterests nclude dstrbuted artfcal ntellgence, dstrbuted smulaton, ntellgent control, power generaton, mult-agent systems, optmzaton technques and advanced dstrbuted computaton. Kwang Y. Lee receved hs B.S. degree n Electrcal Engneerng from Seoul Natonal Unversty, Korea, n 1964, M.S. degree n Electrcal Engneerng from North Dakota State Unversty, Fargo, n 1968, and Ph.D. degree n System Scence from Mchgan State Unversty, East Lansng, n He has been wth Mchgan State, Oregon State, Unv. of Houston, the Pennsylvana State Unversty, and Baylor Unversty where he s currently a Professor and Char of Electrcal and Computer Engneerng. Hs nterests nclude power system control, operaton, plannng, and ntellgent system applcatons to power systems. Dr. Lee s a Fellow of IEEE, an Edtor of IEEE Transactons on Energy Converson, and a former Assocate Edtor of IEEE Transactons on Neural Networks. He s also a regstered Professonal Engneer.