1991), a development of the BLAST program which integrates the building zone energy balance with the system and central plant simulation.

Transcription

1 OPTIMISATION OF MECHANICAL SYSTEMS IN AN INTEGRATED BUILDING ENERGY ANALYSIS PROGRAM: PART I: CONVENTIONAL CENTRAL PLANT EQUIPMENT Russell D. Taylor and Curts O. Pedersen Unversty of Illnos at Urbana- Champagn Dept. of Mechancal and Industral Engneerng W. Green St. Urbana, IL USA ABSTRACT Ths s the frst of two papers that descrbe the development of smulaton methods for optmally controlled central plant equpment whch have been mplemented n the IBLAST (Integrated Buldng Loads Analyss and System Thermodynamcs) buldng energy analyss program. In Part I, conventonal central plant equpment s dscussed, whle Part II covers the smulaton of optmally controlled thermal storage systems. The goal of conventonal central plant optmsaton s to mnmse the operatng cost of the plant equpment. A soluton was acheved by usng a GRG algorthm to mnmse the energy consumpton of each feasble equpment combnaton, for every smulaton tme step, then searchng among these combnatons for the one whch resulted n the least energy cost. Results are presented for several combnatons of conventonal chllng and heatng equpment. INTRODUCTION The purpose of ths research was to use an ntegrated energy analyss program to smulate optmal control of a buldng's central plant equpment. In ths paper control mples the manpulaton of system set ponts, such as the zone supply ar temperature and col water enterng temperature, or the fracton of the load beng met by each central plant component to acheve mnmum energy cost. In a sense, ths s macroscopc control compared to mcroscopc control of the system and plant to adjust valves or dampers or to montor and manpulate duct ar pressures to mantan desred flow rates. The result of the optmsaton, n each case, was an hourly schedule of equpment operaton that mnmsed total energy cost takng nto account both the hourly varatons n energy cost and those assessed accordng to the maxmum plant energy consumpton. Schedulng s crucal to plant optmsaton snce optmsaton of components ndvdually does not generally gve overall optmum performance (Olson, 9, 99, 99, and 99). The buldng smulaton program chosen for ths work was IBLAST (Taylor, 99; Taylor et al, 99, 99), a development of the BLAST program whch ntegrates the buldng zone energy balance wth the system and central plant smulaton. When the cost of energy s constant, mnmsng total energy use s equvalent to mnmsng energy cost. In contrast, ndustral and commercal users of electrcty are usually subject to utlty rates that vary accordng to the tme-of-use, wth ncreased energy costs durng daytme on-peak perods. Addtonally, demand charges put a per klowatt premum on the maxmum power consumpton experenced durng the on-peak hours. Ths type of rate structure requres a control strategy that drectly mnmses energy cost. The optmsaton problem s complcated because there s no drect relatonshp between steady state energy consumpton and peak demand. Generally, the optmal soluton trades ncreased total energy consumpton by operatng at less-than-optmal part load effcency for reductons n peak demand. Ths requres the forecastng of loads to determne the approprate schedule of plant control decsons that mnmses the peak energy consumpton. METHOD The objectve of ths research was acheved by consderng the solutons to two separate optmsaton problems: mnmsng the energy consumpton of the operatng plant equpment at each nstant of the smulaton, and mnmsng the total cost of the energy consumed by the system over a daly cycle. Most large buldngs wth conventonal heatng and coolng equpment, e.g. chllers and bolers, normally have several small unts of each type of equpment connected n parallel rather than one large unt of equvalent capacty. Ths allows the operatng capacty of the plant to be adjusted to optmse the overall plant part load rato. Once the optmal load dstrbuton among the operatng plant components s found for an nstant of tme, the next step s to mnmse the energy cost over an nterval of tme. Ths requres not only that the equpment combnaton change to mnmse total energy consumpton, but also that the peak energy consumpton be factored nto the problem when

2 demand charges are n effect. Dependng on the rate structure, the optmal schedule may requre turnng the addtonal equpment on before the on-peak perod begns, or operatng at a hgher than optmal part load rato rather than turnng on more equpment. When demand charges are beng consdered, the optmsaton problem s one of schedulng the avalable equpment so that t meets the loads as effcently as possble at each nstant of tme, whle avodng spkes n energy consumpton due to start up transents. Consderaton of demand charges n selectng an optmal path through all the possble equpment combnatons complcates the problem substantally as dscussed by Olson (9, 99, 99, and 99). The soluton to the optmal load dstrbuton problem requred fndng the load on each of the operatng plant components whch mnmsed the power consumpton at the nstant the optmsaton was performed. The problem frst requred formulaton of the governng equatons for the heatng and coolng plant. From these equatons, several dfferent types of varables were defned accordng to ther functon: control varables, that are adjusted by the optmser to mprove plant performance; parameters, constant values obtaned from nput; and state varables, that descrbe other aspects of plant performance but are functonally dependent on the controls and the nput parameters. Formulaton of the state equatons was carred out separately for the heatng and coolng plant; however, the objectve functon contaned both heatng and coolng plant energy consumpton nformaton because nteractons between the two types of equpment are common. Next, constrant equatons were defned to ensure that the state and control varables satsfed the fundamental conservaton laws. Fnally, the objectve functon, the quantty beng mnmsed, was formulated. The reader s referred to (Taylor, 99) for addtonal detals on the formulaton of the basc conservaton equatons whch descrbe the ar handlng system and central plant operaton. The objectve functon that was mnmsed, for each plant combnaton at every tme step, was the total cost of the energy consumed durng each smulaton tme step. That s, the power consumpton of each energy type multpled by the approprate cost factor as shown n Equaton : J = C Q elec elec + C Q gas gas + C Q + C Q () boler fuel boler fuel desel fuel desel fuel where J s the objectve functon, C represents the cost factor for each type of energy, and Q s the energy consumpton rate of each energy type. The constrant equaton for the chller plant was conservaton of energy, defned so that the coolng load mnus the contrbuton from each ndvdual chller equalled zero when the constrant was satsfed as shown n Equaton : n chllers c, load Q, coolng = Q = () Equaton s the heatng plant's counterpart to Equaton. Note that waste heat can couple the heatng plant to the coolng plant, as can the heat requred to operate equpment such as, absorpton chllers. nchllers nbolers h, load, waste, heatng = = Q Q Q = () The control varables that were vared to mnmse the energy cost of the plant were the part load ratos for each plant component X,plr as defned by: X Q Q = (), plr, capacty The part load rato, X,plr, must also be constraned between maxmum and mnmum values defned by each pece of equpment. In order to accomplsh the actual optmsaton and determne the best equpment operatng part load ratos at each tme step, a solver was requred. A generalsed reduced gradent (GRG) routne (GRG User's Gude for UNIX, 9) was readly avalable and the problem descrbed above was modfed for soluton by ths method. Implementaton of the GRG solver requred the development of two addtonal subroutnes: one to control the GRG solver and the other to evaluate the objectve functon. However, performng the plant equpment optmsaton n ths manner does not, n general, results n a global mnmum. The solver would have to be allowed to modfy the other parameters affectng plant operaton, such as the plant leavng water temperature, to obtan a true global mnmum. The GRG solver was appled at each tme step, whch could be as short as mnute dependng on smulaton stablty requrements. However, the feasble equpment combnatons were determned on an hourly bass because of the desre to lmt the frequency of changes n operatng plant components and the penaltes assocated wth equpment start-up. As a result, only equpment combnatons whch were feasble for an entre hour were consdered when searchng for the optmal equpment combnaton.

3 It s now approprate to consder the effect demand charges have on the optmal plant operaton schedule. Many utltes mpose demand charges on those customers who consume large amounts of energy n addton to subjectng them to varatons n the energy cost durng a hour cycle. Typcally, demand charges are determned by the peak electrc power consumpton durng the utlty's bllng cycle and s assessed per day of that cycle. For example, Equaton s a smple formula for calculatng the demand charge s gven by: ( demand charge) $ day kw (# days n cycle)( peak kw) = () In practce, the demand charge structure may be consderably more complex and vares between utlty companes. Demand charges penalse peak energy consumpton rates, and n order to avod or at least mnmse these charges, consumers must operate ther equpment so that the largest peaks n power consumpton occur outsde the perod when peak power consumpton s beng tracked. Spkes n power consumpton are most lkely to occur when equpment s turned on because durng the start-up sequence operatng condtons are far from nomnal and addtonal power s requred to accelerate: pumps, fans, motors, etc., up to ther desgn operatng speeds. However, start-up spkes are short n duraton and typcally result n about % hgher power consumpton than steady state operaton (Olson, 9, 99, 99, and 99). Therefore, n order to avod a large demand charge but mantan suffcent capacty to meet the buldng loads, equpment may need to be turned on before the on-peak perod begns. However, the resultng reducton n demand charges can be more than offset because the total amount of power consumed usually ncreases. Clearly, optmsng the plant equpment schedule nvolves a trade-off between ncreased total energy consumpton, because excess plant capacty s mantaned to avod equpment start-up transents, and operatng the equpment at an optmal part load rato but ncurrng many transents as a result of changes n equpment operaton. The ncentve to pursue the former strategy ncreases as the demand charge grows relatve to the other operatng costs. The latter strategy would be the best opton when there s no demand charge snce t guarantees the lowest possble power consumpton durng any and every tme nterval. Up to ths pont the dscusson has looked at mnmsng nstantaneous power consumpton. Now mnmsaton of the total energy cost ncurred over a specfed perod of tme wll be consdered. The soluton to ths type of optmsaton problem requres that the sum of all energy costs over the smulaton perod accountng for dfferent types of fuel, hourly varatons n utlty rates and demand charges be mnmsed. Ths quantty s calculated from Equaton : N fuel types, tot utl. elec peak base ( elec elec ) () () C = cq + c t Q t dt = + c Q Q demand T elec () The frst term on the rght hand sde of ths equaton represents the total cost over the smulaton perod for energy that has a fxed cost per unt of consumpton. That s, the cost of the energy does not vary wth tme of day. The energy use of fossl fuel powered equpment such as desel generators and natural gas fred bolers would be ncluded n ths term. The second term on the rght hand sde of Equaton s used to calculate the total cost of energy when that cost s tme dependent. However, snce utlty electrcty cost normally vares n dscreet amounts, not contnuously, durng the day the second term of Equaton can be wrtten as a summaton, as n Equaton : T () () c t Q t dt = utl. elec N tme steps = elec () () c t Q t t utl. elec elec () The last term on the rght hand sde of Equaton represents utlty demand charges. These are assessed per unt power consumpton at the peak consumpton rate. The dscreet nature of the smulaton n terms of both plant equpment choces and the fnte tme step used to update zone, system, and plant condtons suggests an approach to obtan an optmal equpment schedule whch s llustrated n Fgures, through. In these fgures, each column represents an hour of the smulaton perod. Each block n a column represents the state assocated wth one specfc combnaton of plant equpment. Note that all equpment combnatons may not be feasble durng a specfed hour because of nsuffcent capacty to meet the heatng or coolng loads. For example, n hour

4 combnatons,, and are represented as nfeasble. The lnes from combnatons at one hour to combnatons at the next hour represent changes n equpment usage whch wll, subsequently, be referred to as transtons. Each transton has a base power consumpton rate assocated wth t whch accounts for the current equpment confguraton of the current hour, plus the effects of equpment startup to obtan a new equpment combnaton. Hour Hour Hour Transtons Hour Transtons Hour Fgure : Evaluate All Feasble Equpment Combnatons and Transtons Fgure shows the mnmum energy consumpton path based on the performance of possble equpment combnatons. Ths s determned by selectng, at each hour, the equpment combnaton wth the mnmum energy consumpton. Repeatng ths procedure for every hour gves a sequence of equpment operaton whch results n the mnmum total energy consumpton. Peak, ntal, and fnal power consumptons are gnored ntally. However, these values are needed to determne the demand charge for the mnmum energy path and ultmately allow the total energy cost to be computed. It s not possble, a pror, to determne whether the peak power consumpton results from an equpment transton or the load on the plant equpment. The mnmum energy consumpton path gves a baselne energy cost and peak power consumpton whch are compared to the feasble alternatve plant equpment schedules,.e. sequences of plant combnatons whch dffer from the mnmum energy plant equpment schedule. In order to determne the actual mnmum cost path all the remanng feasble paths must be searched snce the peak power consumpton may result from an equpment transton between one hour and the next. Hour Hour Hour Mnmum Energy Path Fgure : Calculate Mnmum Energy Path Based on Hourly Energy Consumpton However, for a path to have a lower total cost than the mnmum energy path, t must have a lower peak power consumpton because ts total energy consumpton wll, by defnton, be hgher. As a result, equpment combnatons and transtons that have a hgher peak power consumpton than the mnmum energy path may be elmnated, reducng the total number of search paths. Fgure shows how the equpment combnaton and transton elmnatons can substantally reduce the number of search paths requred to fnd the mnmum cost path. Fnally, the remanng allowable paths are searched to determne the mnmum cost path as shown n Fgure. It should be farly obvous that unless the number of search paths can be controlled the optmal soluton rapdly grows beyond the capabltes of most computers to solve n a reasonable perod of tme. Whle the above descrpton s nstructve t does not demonstrate the complextes that occur n plants wth many components. Obvously, the more components the more feasble paths that exst. Olson (9) concluded that an exhaustve search of all the possble paths would be requred to fnd a global mnmum because the peak energy consumpton of a path s a functon of the path tself. Ths concluson led Olson to use heurstc methods to fnd the optmal path because the alternatve, an exhaustve search, was mpractcal. However, the resultng soluton was never guaranteed to be a global mnmum. Wth one caveat, the method descrbed above represents an alternatve to Olson's heurstc approach that does result n a global mnmum energy cost wthout requrng an exhaustve search. The caveat s that whle the peak power consumpton experenced durng equpment start-up s factored nto the determnaton of the optmal path, the addtonal energy consumed as a consequence s neglected. Assumng that chller power requrements are nomnally % hgher than steady state for a perod of. hours then the net ncrease n energy consumed s % over the value for steady state chller operaton for a sngle chller. If other equpment s already

5 operatng when the chller s turned on then ths effect s dluted consderably. Hour Hour Hour Infeasble Combnaton Feasble Combnaton Peak Power Too Hgh Mnmum Energy Path Feasble Transton Infeasble Transton Fgure : Determne Feasble Paths to Search by Elmnatng Combnatons and Transtons Hour Hour Hour Mnmum Energy Path Mnmum Cost Path Fgure : Determne Mnmum Cost Path by Comparng Feasble Paths to Mnmum Energy Path Equaton shows that an arbtrary ntal path has costs derved from both total energy consumpton and peak consumpton. A good choce for the ntal path s the mnmum energy path because t can be determned easly. The total cost assocated wth ths path s: mn C = C Q + C Q mn ref tot peak () If the demand charge C s zero then ths s the least cost path. However, f C s non zero then ths path may or may not also be the mnmum cost path. The total cost of any other arbtrary but feasble path s gven by Equaton 9: C = C Q + C Q tot peak (9) But, for any other path, the total energy consumed must be greater than the energy consumed by the least cost path. Ths s gven by Equaton : Q tot > Q mn () tot Therefore, for the total cost of path C to be less than Cref, Equaton must hold: Q peak mn peak < Q () Ths means that any path segment, or transton, between equpment confguratons havng a hgher peak consumpton than the reference path can be elmnated from further consderaton. The global mnmum may thus be found by searchng on a much reduced subset of the feasble paths. RESULTS In order to demonstrate the feasblty of the optmsaton scheme descrbed above, a central plant consstng of two kw electrc powered chllers, one kw electrc powered chller, one kw desel powered chller, and one kw electrc boler to supply domestc hot water was specfed to serve a two zone buldng wth a VAV system. The nfluences of the relatve cost of each type of energy and the demand charge on the least cost equpment schedule were of most nterest. In the frst set of examples, the demand charge was /kw and the on-peak rate multpler was. Desel cost was gven values of./kwh, /kwh, /kwh, and /kwh to demonstrate how the mnmum cost schedule would change wth ths parameter. Fgure shows the mnmum energy equpment schedule when the cost of desel fuel s./kwh. Ths schedule does not utlse the desel chller at all and the electrc boler s used to supply domestc hot water. Fgure, the mnmum cost schedule, provdes a sgnfcant contrast snce the desel chller operates durng hours through and the boler does not run because waste heat from the desel chller meets the domestc hot water load. However, durng hour the desel chller produced nsuffcent waste heat and the boler operated at a mnmal part load fracton. Durng hour, a transton causes all the chllers to operate and much more capacty s avalable than for the mnmum energy schedule. In ths stuaton, the peak electrc consumpton s a steady state peak and s not due to an equpment start-up transent. Further analyss ndcated that turnng on the desel chller shfted the peak electrc consumpton to hour, n the mnmum cost schedule, from hour, n the mnmum energy schedule. In addton, a sgnfcant reducton n peak electrc consumpton occurred because the desel chller allowed the electrc powered chllers to operate at lower part load fractons. Fgure shows the hour-by-hour dfference n total energy consumpton between the mnmum energy and mnmum cost schedules represented n Fgures

6 and. The least cost schedule s entrely the result of the demand charge, otherwse the total cost of the energy consumed by the mnmum energy path s lower than the total energy cost of the mnmum cost path. mnmum cost path uses the desel chller durng the on-peak hours, as shown n Fgure. However, when the cost of desel fuel s greater than ths value, the total cost, ncludng demand charge, for usng all electrc chllers becomes less than for usng a combnaton of desel and electrc chllers, even wth the reducton n peak electrcty demand. Operatng Capacty (kw) Chller Desel Chller Boler Cost Dfferental 9 9 Fgure : Mnmum "Energy" Plant Operatng Schedule for a Combnaton of Electrc and Desel Chllers and an Electrc Boler (Desel Cost =./kwh) Desel Fuel Cost Fgure : Total Cost Dfferental Between Mnmum "Energy" and Mnmum Cost Schedules as Desel Fuel Cost s Increased Operatng Capacty (kw) Chller Desel Chller Boler Operatng Capacty (kw) Chller Desel Chller Boler 9 9 Fgure : Mnmum Cost Plant Operatng Schedule for a Combnaton of Electrc and Desel Chllers and an Electrc Boler (Desel Cost =./kwh) Energy Consumpton (kw) Mn Energy Mn Cost 9 9 Fgure : Dfference n Actual Energy Consumpton Between Mnmum Energy and Mnmum Cost Schedules (Desel Cost =./kwh) Fgure shows how ncreasng the cost of desel fuel affects the dfference n total cost between the mnmum cost and mnmum energy schedules. When desel costs less than about./kwh the 9 9 Fgure 9: Mnmum "Energy" Plant Operatng Schedule for a Combnaton of Electrc and Desel Chllers and a Fuel Fred Boler (Desel Cost =./kwh) A second example was generated by replacng the electrc boler wth a fuel fred boler of the same capacty. In addton, a desel fuel cost of./kwh and a boler fuel cost of./kwh were specfed. Fgure 9 shows that the mnmum energy equpment schedule for ths case s dentcal to the plant schedule shown n Fgure. However, the mnmum cost plant schedule, shown n Fgure, s sgnfcantly dfferent from the correspondng fgure of the prevous example, Fgure. The mnmum cost path n ths example stll utlsed the fuel boler for hour but the remanng electrc chller was not turned on at the end of hour, as n the prevous case. In the prevous example, the peak electrc consumpton occurred at hour for mnmum cost path and the electrc boler was turned on, contrbutng to the peak electrcty load. In ths example, the boler s fuel fred so that, although the

7 peak electrc consumpton also occurs at hour, t s a lower peak. Turnng on the chller at hour, as n the prevous example, would ncrease peak electrc consumpton and move the tme at whch t occurs to hour. Not turnng on the addtonal chller and operatng all the equpment at overall hgher part load fractons to meet the coolng load avoded ths peak n consumpton. Cost Dfferental On-Peak Rate= On-Peak Rate= On-Peak Rate= On-Peak Rate= Operatng Capacty (kw) Chller Desel Chller Fuel Boler Demand Charge Fgure : Effect of Demand Charge and On-Peak Electrc Rate On the Dfference Between Mnmum Energy and Mnmum Cost Schedules (Desel Cost =./kwh) 9 9 Fgure : Mnmum Cost Plant Operatng Schedule for a Combnaton of Electrc and Desel Chllers and a Fuel Fred Boler (Desel Cost =./kwh) The equpment from the prevous example was also used to evaluate the senstvty of the mnmum energy and mnmum cost equpment schedules to varatons n demand charge and on-peak electrcty rate multpler. The demand charge was vared from /kw to /kw for a fxed value of the on-peak rate multpler and four on-peak electrc rate multpler values were used:,,, and. The results are gven n Fgure, showng the addtonal cost of the mnmum energy schedule compared to the mnmum cost plant operatng schedule. As the demand charge was ncreased, n all cases except for an on-peak rate multpler of, the dfference n cost between the two schedules ncreased. The mnmum energy and mnmum cost paths also converged when the demand charge was reduced to less than /kw. These results matched those obtaned n prevous cases except when the onpeak electrc rate multpler was. An example of ths case s plotted n Fgure for a demand charge of /kw, showng the on-peak use of desel chllers because, per unt coolng, electrcty was more expensve than desel fuel. Another effect was the rasng of the desel chller part load rato at hour so that the boler was not requred to supply any of the domestc hot water load. Operatng Capacty (kw) Chller Desel Chller Boler 9 9 Fgure : Optmal Plant Equpment Schedule when On-Peak Electrcty = /kwh, Demand Charge = /kw and Desel =./kwh CONCLUSIONS In order to determne optmal plant equpment schedules, the feasble optmal equpment combnatons at each tme step had to be evaluated. The optmsaton was based on selectng the part load rato at whch each pece of avalable equpment was to be operated. However, when more than one equpment type was used, and especally when the dfferent types were powered by dfferent energy sources, the optmal soluton was strongly dependent on the relatve dfference between the energy rates and, when applcable, ther varatons durng a one day cycle. The optmal equpment schedules descrbed n ths research were determned by pecng together, for each hour of the smulaton, a seres of equpment combnatons whose energy cost had been mnmsed by selectng the approprate part load ratos for each pece of equpment. Normally, n order to fnd the lowest cost schedule, a search would have to be conducted on all the feasble paths resultng n the total number of possble paths ncreasng untl

8 requred computaton tme and data storage became prohbtvely large. An alternatve to a global search of all possble paths was used n solvng ths problem. Ths alternatve recognsed that only a lmted subset of the possble paths need be consdered as canddates for a true global mnmum. Ths subset was made up of those paths that had lower peak energy consumpton than the reference mnmum energy. In cases where the number of possble equpment combnatons was large, the reducton n the total number of paths calculated was typcally several orders of magntude. Furthermore, the method fnds a true global mnmum energy cost under the assumpton that the cost of the addtonal energy consumed durng an equpment start-up spke s neglgble. The optmal schedulng method developed n ths paper provded results that follow expected trends. For example, as the utlty demand charges ncrease, the smulaton model predcts that fewer changes n equpment operatng capacty should occur. However, the results do not lead to obvous generalsatons or rules-of-thumb. The wde selecton n equpment types and the varatons n energy costs mply that a smulaton of the proposed buldng and ts equpment s necessary to accurately determne the best way to operate equpment. REFERENCES GRG Users Gude for UNIX, Computng Servces Offce, Unversty of Illnos at Urbana- Champagn, January 9. Olson, R.T., Optmal Allocaton of Buldng Coolng Loads to Chlled Water Plant Equpment, Ph.D. Thess Unversty of Illnos at Urbana-Champagn, 9. Olson, R.T. and J.S. Lebman, "Optmsaton of a Chlled Water Plant Usng Sequental Quadratc Programmng", Engneerng Optmsaton, 99, Vol., pp. -9. Olson, R.T., C.O. Pedersen and J.S. Lebman, "A Dynamc Procedure for the Optmal Sequencng of Plant Equpment Part I: Algorthm Fundamentals", Engneerng Optmsaton, 99, Vol. (), pp. -. Control, Ph.D. Thess, Unversty of Illnos at Urbana-Champagn, May 99. Taylor, R. D. C.O. Pedersen, and L. Lawre, "Smultaneous Smulaton of Buldngs and Mechancal Systems n Heat Balance Based Energy Analyss Programs," Proceedngs of the rd Internatonal Conference on System Smulaton n Buldngs, Lege, Belgum, December -, 99, pp.. Taylor, R. D., C.O. Pedersen, R.J. Lesen, D. Fsher, and L. Lawre, "Impact of Smultaneous Smulaton of Buldngs and Mechancal Systems n Heat Balance Based Energy Analyss Programs on System Response and Control," IBPSA Buldng Smulaton '9: nd Internatonal Conference Proceedngs, Nce, France, August -, 99, pp.. NOMENCLATURE Varables c demand - demand charge cost factor c utl, elec - utlty electrc cost factor t - tme C - Cost coeffcent J - Value of the objectve functon Q tot - Total energy consumpton of a path Q - Power consumpton Q peak - Peak power consumpton along a path Q base elec - Power consumpton threshold for calculatng demand charge Q peak elec - Peak power consumpton for calculatng demand charge T - temperature X - Equpment part load rato Superscrpts mn - value assocated wth the mnmum energy path Subscrpts peak - maxmum assocated wth a path ref - assocated wth a reference path Olson, R.T. and J.S. Lebman, "A Dynamc Procedure for the Optmal Sequencng of Plant Equpment Part II: Valdaton and Senstvty Analyss", Engneerng Optmsaton, 99, Vol., pp. -. Taylor, R. D., Development of an Integrated Buldng Energy Smulaton wth Optmal Central Plant