Model-Based Predictive Control of a Multi-Evaporator Vapor Compression Cooling Cycle

Transcription

1 28 America Cotrol Coferece Westi Seattle Hotel, Seattle, Washigto, USA Jue 11-13, 28 WeC7. Model-Based Predictive Cotrol of a Multi-Evaporator Vapor Compressio Coolig Ccle Matthew S. Elliott ad Bra P. Rasmusse Abstract This paper presets a decetralized cotrol architecture for multiple evaporator vapor compressio sstems usig model-based predictive cotrol. Vapor compressio sstems are widel used for heatig, air-coditioig ad refrigeratio, ad costitute a maor part of total US eerg use. Advaced cotrol strategies have the potetial to sigificatl icrease eerg efficiec, while deliverig the ecessar amout of coolig capacit. This paper proposes a decetralized cotrol approach based o a stud of iteractig damics, wherei the coolig capacit of each evaporator is cotrolled b a multi-iput, multi-output MPC cotroller ad stadard PI cotrollers are used to regulate sstem pressures b modulatig compressor speed ad discharge valve opeig. This is i cotrast with traditioal sigle-iput, sigle-output cotrol approaches, which ca result i udesired damic behavior. The efficac of the proposed cotrol architecture is demostrated o a experimetal sstem. I. INTRODUCTION Vapor compressio coolig (VCC) ccles are the primar meas of mechaical coolig toda; the ca be foud i settigs ragig from household refrigerators to office buildigs. Cotrol techiques for these ccles have traditioall cosisted of simple electromechaical devices ad o/off cotrol strategies. Techological advaces such as variable speed compressors, electroicall cotrolled expasio valves, ad improved computig speed ow allow more precise ad efficiet cotrol of these ccles. Sice world eerg demad cotiues to icrease, ad air coditioig is a maor compoet of that demad, the adaptatio of advaced cotrol strategies to VCC ccles has the potetial to make a serious impact o eerg cosumptio. This paper presets a method of cotrollig a multiple evaporator sstem with a ovel cotrol architecture that uses a decetralized approach. The coolig capacit of each evaporator is regulated b its ow multiple-iput, multiple output (MIMO) cotroller, while the compressor ad discharge valves use sigle-iput, sigle-output (SISO) PI cotrollers to regulate sstem pressures. The MIMO loops use a model-based predictive cotroller that takes desired coolig capacit ad evaporator superheat as the regulated outputs ad the expasio valve opeig ad water flow valve opeig as cotrollable iputs. This decetralized architecture reflects the spatiall distributed Matthew S. Elliott is a graduate researcher with the Texas A&M Uiversit Departmet of Mechaical Egieerig, College Statio, TX. Bra P. Rasmusse is a Assistat Professor of Mechaical Egieerig at Texas A&M Uiversit, College Statio, TX. (phoe: , fax: , brasmusse@tamu.edu) ature of the phsical sstem, while esurig efficiec ad performace demads are met. Fig. 1: Geeral Multi-Evaporator Sstem with desigated states. II. BACKGROUND ON VAPOR COMPRESSION CYCLES A. Geeral Multiple Evaporator Sstem I this paper we cosider the geeral multi-evaporator vapor compressio ccle show i Figure 1 ad adapted from the sigle evaporator sstem detailed i [4]. The first stage of the thermodamic ccle is at the ilet of the compressor, where refrigerat is i a low pressure, gaseous state (deoted as state 1 i Figure 1). The compressor adds eerg to the fluid b compressig it to a high pressure, high temperature gas (state 2). This gas passes ito the codeser, where heat eerg is reected from the refrigerat to the secodar fluid (water or air). This causes the refrigerat to codese to a high pressure liquid. A receiver at the ed of the codeser esures that the refrigerat becomes a saturated liquid (state 3). This saturated liquid is fed ito a set of expasio valves, which meter the /8/$2. 28 AACC. 1463

2 refrigerat flowig ito the evaporators. The refrigerat is ow a two-phase fluid (states 4 ad ). This two phase fluid absorbs heat from the water eterig the evaporators, chillig the water ad causig the refrigerat to evaporate. This low pressure gas exits the evaporators ad returs to the compressor. The discharge valve (SDR) o the secodar evaporators creates a pressure differetial betwee evaporators, thus allowig them to provide coolig at differet saturatio temperatures. Multiple evaporator sstems allow differet amouts of coolig at differet temperatures to be delivered to differet regios i the same overall sstem, such as apartmet uits or large office buildigs. These sstems ca also allow for storig perishables requirig differet storage temperatures, such as i a supermarket case or i a refrigerated trailer truck. Sice the evaporator temperature must be lower tha the temperature of the fluid beig cooled, pressure is a importat operatig coditio of the sstem ad must be cotrolled. However, from the stadpoit of overall sstem performace, evaporator coolig is paramout. If the evaporators are ot removig sufficiet heat from the secodar fluid, the sstem is ot meetig its performace requiremets, regardless of the evaporator temperature. The fial importat elemet for cotrol of the sstem is superheat, which occurs whe the temperature of the refrigerat at the exit of the evaporator is higher tha the saturatio temperature at evaporator pressure. The existece of superheat is a guaratee that the evaporator is experiecig complete evaporatio of the refrigerat; if superheat is lost, liquid refrigerat ca pass ito the compressor, which ca damage it. However, excessive superheat meas that the evaporator is operatig iefficietl, sice the amout of coolig occurrig drops sigificatl oce the fluid is completel evaporated. Superheat ca be icreased b icreasig compressor speed, closig the expasio valve, or opeig the discharge valve, sice all of these actios lower the pressure, ad therefore the saturatio temperature, of the evaporator. Icreasig the water flow across the valve also icreases superheat, sice it icreases heat trasfer from the refrigerat. Cotrollig these three coditios coolig, pressure, ad superheat i each evaporator cotrols the operatio of the sstem ad therefore provides a meas to deliver the required coolig i the most efficiet wa possible. I traditioal sigle evaporator sstems, a mechaical expasio valve is used to regulate superheat. If the compressor is variable speed or two-stage, its speed is icreased to meet icreased coolig demad; sice this icreases superheat, the valve opes to reduce superheat to the desired level. I a multi-evaporator sstem, the damics of the two evaporators are ver tightl coupled; chages i oe expasio valve have a strog effect o the superheat ad pressure of the other evaporators. Therefore, a completel SISO cotrol approach will lead to oscillator or dagerous limit ccle behavior. While this could potetiall be addressed with a large, cetralized MIMO cotroller that rus the etire sstem, this sort of cotrol is difficult to implemet, give the spatiall distributed ature of idustrial sstems ad the complexit of the damics ivolved. Therefore, a more decetralized approach is preferred; the desired arragemet will give each evaporator its ow cotroller that operates idepedetl of the rest of the sstem but does ot coflict with the obectives of the eighborig compoets. B. Dual Evaporator Experimetal Sstem For the research detailed i this paper, a two-evaporator water chiller test apparatus was used. This test apparatus has variable cotrol of the electroic expasio valves (EEVs), compressor, ad water flow valves (WFVs). Furthermore, the secod evaporator has a discharge valve (SDR) that allows the two evaporators to fuctio at differet pressures. Usig multiple evaporators with EEVs allows differet amouts of coolig to be delivered to differet regios at differet temperatures, such as differet uits i a apartmet buildig. The refrigerat used is R134a, which is a HCFC widel used i automotive ad idustrial sstems. I order to measure sstem properties, icludig regulated variables, trasducers are placed at saliet poits of the thermodamic ccle. Pressure trasducers are placed at the outlet of each evaporator ad the codeser to measure the saturatio pressures of the refrigerat. Thermocouples are immersed i the refrigerat flow at the ilet ad outlet of each evaporator ad the codeser. These temperature ad pressure measuremets allow computatio of fluid properties such as ethalp ad desit at the relevat poits of the ccle. I order to measure mass flow of refrigerat through each evaporator, the calculated fluid desities are used i couctio with turbie-tpe volumetric flow meters placed at the ilet of each EEV. These measuremets allow o-lie computatio of coolig ad superheat of each evaporator. Coolig is measured as: ( h h ) & = & (1) Q m ref, e, o e, i Q & & m ref, Coolig at the th evaporator, kw Mass flow of R134a for th evaporator, kg/sec h e, Ethalp of R134a at th evaporator exit, kj/kg o h e, Ethalp of R134a at th evaporator ilet, kj/kg i For this cotroller, superheat is calculated as: SH = Te, ro T (2) sat, e SH Superheat of the th evaporator, C T e, Exit R134a temperature of th evaporator, C ro T sat, R134a saturatio temperature at th evaporator e pressure, C 1464

3 III. DYNAMIC ANALYSIS The first step towards adaptatio of advaced cotrol to this VCC ccle is developig a uderstadig of the relatioships betwee cotrol iputs (e.g., compressor speed, EEV opeig) ad outputs (e.g., evaporator pressure, coolig), as well as the iterdepedece betwee evaporators. I order to develop this uderstadig, a set of experimetall derived liear models were created at selected operatig coditios [4]. The sstem was allowed to come to stead state operatio, ad the excited with a pseudo-radom biar iput. Because of the differig uits ad disparate scalig, the iputs ad outputs were ormalized before costructig empirical models, allowig the effects of each iput ito each output to be compared accuratel [3]. The models are 1 st, 2 d, ad 3 rd order predictio-error method (PEM) state-space models, crossvalidated to esure accurac. Figure 2 shows the frequec resposes of the models derived b steppig the compressor speed. SISO models were developed for RPM to each output; the DC gai ad badwidth are show. Clearl, compressor speed has a strog effect o evaporator pressures ad superheats. This effect is stroger at lower speeds. However, varig the compressor speed aloe has a weak effect o coolig; if the valve does ot ope to permit additioal refrigerat mass flow, icreasig compressor speed has the effect of icreasig codeser pressure, ad the additioal eerg iput to the sstem b the compressor is reected ito the codeser. This suggests that the compressor ca be used to deliver the eerg ito the sstem ecessar to perform the coolig work without chagig the coolig itself. Figure 3 shows the effects of the first EEV. The EEV has a strog effect o the pressure ad superheat of both evaporators at all operatig coditios. The effect of the EEV o the first evaporator s coolig is also strog; however, the effect of the first EEV o the secod evaporator s coolig is oexistet. Figure 4 details the resposes to chages i the SDR. It has a strog effect o the pressure of the secod evaporator, ad therefore its superheat. It does ot affect the first evaporator at stead state, which suggests that cotrol of the first evaporator ca be separated from the SDR. Fiall, Figure details how the sstem respods to chages i water flow of the first evaporator via step chages i WFV #1. It has a strog effect o superheat ad a smaller effect o coolig, ad does ot have a effects o the secod evaporator Evap 1 Coolig Evap 2 Coolig - Evap 2 Coolig - - Evap 1 Coolig Fig 2: Normalized Frequec resposes to step chages i RPM Evap 2 Coolig Evap 1 Coolig Fig 3: Normalized Frequec resposes to step chages i EEV Fig. 4: Normalized Frequec resposes to step chages i SDR Evap 1 Coolig Evap 2 Coolig Fig. : Normalized Frequec resposes to step chages i WFV1. 146

4 IV. PROPOSED CONTROL ARCHITECTURE I the proposed cotrol structure, the expasio valves are used primaril to regulate coolig, sice the cotrol how much refrigerat eters the evaporator, ad coolig is strogl depedet o mass flow. Their actios have a strog iput o superheat as well. The water flow valves will be used to help regulate superheat. These two actuators ca be coupled to regulate coolig (the primar cotrol obective) ad superheat, resultig i a 2-iput, 2-output plat for each evaporator. This is differet from the idustriall stadard approach of usig EEVs to cotrol evaporator superheat. Figure 6 shows block diagrams of the proposed architecture. + - Q1 Q2 SH1 SH2 Evap 1 MPC 1 D P evap Evap 2 MPC 2 PI 1 Compressor EEV1 WFV1 Evap 1 Plat ad Cotroller Evap 2 Pressure PI 2 SDR EEV2 WFV2 Evap 2 Plat ad Cotroller Figure 6: Proposed Cotrol Architecture RPM % Ope Whe cotrollig superheat, the critical requiremet is that superheat is preset. Oce this coditio is met, the primar advatage of a close regulatio of superheat is that as superheat is drive dow to a miimum value, the evaporator operates with icreasig efficiec. This regulatio has traditioall bee performed with the EEV. I the proposed architecture, coolig ad evaporator pressure regulatio ca be performed regardless of the amout of superheat. From the perspective of coolig ad pressure cotrol, strict regulatio of superheat to a setpoit is uecessar; it is ol ecessar to esure that it stas above a miimum value to protect the compressor, but withi a reasoable bad so that the evaporator does ot operate iefficietl. Therefore, a cotrol strateg is required that respects the phsical limits of the actuators, will regulate coolig to a specific value, ad will keep superheat withi a defied bad of operatig coditios without exertig uecessar cotroller effort. This leads aturall to adoptig a model predictive cotrol approach. The term model predictive cotrol (MPC) refers to a suite of cotrol strategies origiall developed i idustr durig the 197s. These approaches all use a explicit model of the phsical sstem to derive the set of cotroller actios that miimize a cost fuctio subect to a set of costraits. At each samplig istat, the cotroller calculates the cost over the predictio horizo, ad selects the cotrol actios over the cotrol actio horizo that miimizes the user-defied cost fuctio. It applies the first of these cotrol actios, ad the at the ext samplig istat repeats the process. I this wa the predictio horizo recedes at each samplig istat; hece, MPC is also kow as recedig horizo cotrol. Oe of the great advatages of MPC is its iheret abilit to accout for costraits. These costraits ca be iheret to the actuators, e.g., a valve ca ot ope past 1% ope or close past % ope. While classical cotrol techiques like PID loops ca be modified with a saturatio to esure that actuator limits are ot exceeded, MPC has the advatage of beig able to foresee ad pla for these limitatios, which ca improve sstem performace over the log term [1]. MPC also has the advatage that additioal costraits ca be defied b the user to keep the sstem operatig i a safe rage, e.g., keepig evaporator superheat above a desired miimum. MPC has bee adapted to HVAC sstems as a sstem goveror ([8], [9]), ad used to cotrol coolig of a sigle evaporator sstem [7]. Oce the evaporator cotroller is implemeted, the evaporators eed a wa to commuicate to the compressor how much eerg iput is eeded i order to achieve the total desired coolig. Sice the compressor has a large impact o evaporator pressure, ad chages i the expasio valve to meet coolig demad chage the evaporator pressure, pressure of the first evaporator is chose as the sigal to commuicate this eed to the compressor. The first evaporator pressure is chose because the pressure differetial betwee the two evaporators ca be idepedetl regulated b the SDR valve. With this approach, the pressure of the two evaporators ca be regulated usig two SISO PI loops. For example, as oe of the EEVs opes to allow more mass flow ad achieve the set coolig capacit, the pressure of the evaporator will rise. The compressor will speed up to drop the pressure. Meawhile, the pressure differetial betwee the two evaporators is regulated b the SDR i the secod evaporator. Therefore, 6 outputs two evaporator pressures, two amouts of coolig, ad two superheats are regulated usig two 2x2 MIMO plats ad two SISO PI loops. Oe of the great advatages of this decetralized approach is that it is expadable to a large umber of evaporators etworked together over large phsical distaces, without the eed for ureasoable icreases i computig power. 1466

5 V. IMPLEMENTED CONTROLLER PERFORMANCE The proposed architecture was implemeted o the experimetal sstem, ad was successful i regulatig coolig ad pressures, ad keepig superheat above a miimum level. The PI loop cotrollig the compressor to regulate the pressure of the first evaporator has proportioal ad itegral gais of K P =.4 ad K I =.3. The PI usig the SDR to regulate the pressure differece betwee evaporators has gais of K P =.2 ad K I =.. These gais were developed usig a Ziegler-Nichols tuig algorithm as detailed i []. The MPC cotroller was implemeted usig the MatLab MPC toolbox. The cotroller miimizes the followig cost fuctios as detailed i [6]: P { [ ]} 2 ( k + i) ( k) = w r ( k + i) S (3) Δu i= 1 = 1 M { } 2 mv Δu ( k) = w Δu ( k + i 1) S (4) i= 1 = 1 Equatio (3) computes the weighted sum of squared deviatios. This portio of the cost calculatio pealizes deviatio of the outputs from the setpoits. k Curret samplig iterval k + i Future samplig iterval Predictio horizo P w r Number of plat outputs Weight of output ( k + i) ( k i) predicted deviatio at istat k+i + Equatio (4) computes the weighted sum of cotroller adustmets. This equatio pealizes large chages i the actuators, ad makes the cotroller more robust. M Cotrol horizo k + i Future samplig iterval P Predictio horizo Number of maipulated variables (iputs) mv u w Δ Δ u Weight of chage i iput ( k + i 1) predicted adustmet of iput u at future istat k+i-1 I the MPC cotrollers, weights of 1 ad (i equatio (3)) were placed o the coolig ad superheat, ad rate weights of 1. ad.1 (equatio (4)) were placed o the EEV ad WFV, respectivel. A cotrol iterval of 2 secods was used, with a cotrol horizo of 3 itervals (6 secods) ad a predictio horizo of 3 itervals (7 secods). The EEVs were costraied betwee 7% ad 4%; the WVFs were costraied betwee 2% ad 6%. A output costrait was placed o superheat boudig it betwee 3 C ad 2 C. Sice o weight was attached to superheat, the cotroller will ot take steps to regulate it uless the model predicts that superheat will exceed the specified costraits. The plat model used is a idetified secod order PEM state space model with EEV ad WFV positio as the iputs ad superheat ad coolig as the outputs. A experimetal ru to verif setpoit trackig of the idividual local cotrollers was performed; figures 7-9 show the cotroller performace for this experimetal ru. These figures show that setpoit trackig for this cotrol architecture is achieved without the cotrollers fightig each other, which idicates that if a global cotrol law ca be developed to determie the most eerg efficiet pressure ad coolig setpoits, the local cotrollers will be able to meet the global law s requiremets. Note that this test ru cosists of trackig of radoml selected setpoits ad is ot iteded to achieve optimal eerg efficiec. Figure 7 shows the coolig setpoits ad actual coolig vs. time for a experimetal ru. Figure 8 shows that superheat is maitaied above 1 C without goig higher tha 2 C. Superheat ever actuall approaches the lower boud i this ru; this is because the combiatio of pressure ad coolig setpoits is ot the most eerg efficiet combiatio for the sstem coditios. A higher pressure at the coolig setpoit would allow the compressor to ru at a lower speed for the same amout of coolig, resultig i higher efficiec. As more coolig is requested of either valve, it opes to permit the extra mass flow eeded. Sice this has the effect of icreasig evaporator pressures, the compressor speed icreases to keep pressure at the required setpoit; the et effect is that as more coolig is eeded, the compressor icreases the eerg iput to the sstem i order to achieve the desired coolig. For a decrease i coolig requiremets, the reverse process occurs. Figure 9 shows the evaporator pressures durig the same test ru. Durig the large chage i coolig setpoits at approximatel 7 ad 72 secods, the pressure rises, ad the compressor speed icreases to drive it back towards its setpoit. The compressor respose is see i Figure 1. Coolig (kw) Evap 1 Coolig ad Setpoit Evap 2 Coolig ad Setpoit Fig. 7: Setpoit trackig of Coolig 1467

6 Superheat ( C) Fig. 8: Regulatio of Evaporator Superheat for MIMO MPC Pressure (kpa) ad Set VI. FUTURE WORK The ext step i this effort is to use the successful compoet cotrollers i coordiatio with a global cotroller that maximizes the eerg efficiec of the water chiller sstem. Sice the ultimate goal of this research is to ope aveues for miimizig eerg cosumptio, this global cotroller would act i a supervisor role to regulate the chilled water temperature i the most efficiet maer possible, settig the desired pressures ad coolig of each of the compoets, ad drivig superheat to a miimum costrait. The abilit of MPC to explicitl accout for costraits allows MPC-cotrolled processes to operate ear those costraits; i this case, the evaporators will be drive to operate much more closel to a miimum level of superheat tha would ormall be advisable [1]. This approach ca help establish a level of efficiec greater tha that achieved b simpl alterig the setpoits. The MPC framework also allows for demad sheddig, where a eerg cosumptio costrait ca be imposed ad coolig will ot meet its setpoit uless the costrait is met. Aother importat frot will be to accout for oliearities i the sstem; as curretl implemeted, the cotrollers are detued such that the fuctio over a wide rage of coditios at a sacrifice of performace. Usig models that more accuratel predict the behavior at curret coditios will allow the cotroller to be more aggressive i setpoit trackig. 2 ad Set Fig. 9: Refrigerat Pressures Compressor Speed (RPM) Respose to chages i demad Compressor ramps up as coolig demad icreases REFERENCES [1] E.F. Camacho ad C. Bordos, Model Predictive Cotrol. Lodo: Spriger, [2] S. Skogestad ad I. Postlethwaite, Multiple Feedback Cotrol: Aalsis ad Desig, 2 d ed. West Sussex, Eglad: Joh Wile & Sos, Ltd., 2. [3] Leart Lug, Sstem Idetificatio: Theor for the User, 2 d ed. Upper Saddle River, New Jerse: Pretice Hall PTR, [4] M. J. Mora ad H. N. Shapiro, Fudametals of Egieerig Thermodamics, 3 rd ed. New York: Joh Wile & Sos, [] K. Ogata, Moder Cotrol Egieerig, 4 th ed. Upper Saddle River, New Jerse: Pretice Hall Ic., 22. [6] MATLAB Model Predictive Cotrol Toolbox Documetatio: [7] D. Leducq, J. Guilpart, G. Trstam, No-liear predictive cotrol of a vapour compressio ccle, Iteratioal Joural of Refrigeratio, vol. 29, pp , August 26. [8] M. Xu, S. Li, W. Cai, L. Lu. Effects of a GPC-PID cotrol strateg with hierarchical structure for a coolig coil uit, Eerg Coversio ad Maagemet, vol. 46, pp , Jauar 26. [9] J. MacArthur. Recedig Horizo Cotrol: A model-based polic for HVAC applicatios, Proceedigs of the 1993 Witer Meetig of ASHRAE Trasactios, Jauar Fig. 1: Compressor Speed Regulatig Evaporator Pressure 1468