Nonlinear superheat and capacity control of a refrigeration plant

Transcription

1 Nonlinear suerheat and caacity control of a refrigeration lant Henrik Rasmussen Deartment of Electronic Systems Aalborg University DK-92 Aalborg, Denmark hr@es.aau.dk Lars F. S. Larsen Danfoss A/S, Nordborg, Denmark Lars.Larsen@Danfoss.com Abstract This aer rooses a novel method for suerheat and caacity control of refrigeration systems. A new low order nonlinear model of the evaorator is develoed and used in a backsteing design of a nonlinear controller. The stability of the roosed method is validated theoretically by Lyaunov analysis and exerimental results shows the erformance of the system for a wide range of oerating oints. The method is comared to a conventional method based on a thermostatic suerheat controller. NOMENCLATURE L e l e ṁ e h i h g h o h lg T e f com T SH T w,in T w,out ṁ w c w α w α e B H time derivative oerator d/ length of the evaorator length of the evaorator two hase section refrigerant mass flow rate secific enthaly, inlet evaorator secific enthaly, end of two hase section evaorator secific enthaly, outlet evaorator secific evaoration energy, refrigerant refrigerant boiling temerature refrigerant ressure, evaorator comressor seed suerheat, evaorator temerature of water into the evaorator temerature of water out of the evaorator mass flow of water secific heat caacity of water heat transfer coefficient water-wall heat transfer coefficient refrigerant-wall wih of evaorator hight of evaorator secific volumen I. INTRODUCTION The basic comonents in a refrigeration system are exansion valve, evaorator, comressor and condenser. One of the key variables that greatly effects the efficiency of the system, is the filling of the evaorator. The filling is indirectly measured by the suerheat defined as the difference between the outlet temerature of the gas and the evaoration temerature. Conventionally the suerheat is controlled by adjusting the oening degree of the exansion valve. To utilize the otential of the evaorator to its maximum the filling should be as high as ossible, i.e. the suerheat should be ket as low as ossible. This is a common control strategy and examles can be found in [8], [7] and [9]. However the fact that the suerheat is highly nonlinear deended on the oint of oeration, the evaorator design and the characteristic of the exansion valve, limits the obtainable erformance with standard PID controllers. Heat load Q & Fig. 1. loos. load Tcr Pum T oe Evaorator Suction ressure control N C Comressor Suerheat control OD Exansion Valve Surroundings Condenser m& ref T ic P c T oc Condenser ressure control Layout of the test refrigeration system including conventional control Previously work by [4] and [6] has roved that gain scheduling is a way to handle these gain variations. In a refrigeration system with variable seed comressor controlling the suction ressure and an exansion valve controlling the suerheat the effect of cross couling between the loos (hunting) may lead to instability or unaccetable erformance, as described in [11]. Motivated by these difficulties, this aer rooses a novel aroach to a model-based suerheat and caacity control. As for the conventional controller the new control strategy controls the suerheat temerature by the the oening degree of the exansion valve and the suction ressure by the comressor seed. Based on a develloed low order nonlinear model, with refrigerant flow and comressor seed as inut and suerheat temerature and suction ressure as outut, a method based on backsteing is used for the controller design. Because backsteing design is based on Lyaunov stability, the controller is stable with a nerly erfect decoubling between caacity and suerheat temerature for resonable coice of gains in the controller. Exeriments on a test system shows an excellent erformance both during startu and for variation of coling caacity by ste change of the comressor seed between minimum and maximum. The new controller is also comared to a conventional controller based on a thermostatic exansion valve (TXV) for controlling of the suerheat. W & C T aoc II. SYSTEM DESCRIPTION The test system fig. 1 is a simle refrigeration system with water circulating through the evaorator. The heat load on the system is maintained by an electrical water heater N CF Fan T a W & CF

2 with an adjustable ower suly for the heating element. The comressor, the evaorator fan and the condenser um are equied with variable seed drives so that the rotational seed can be adjusted continuously. The system is furthermore equied with an electronic exansion valve that enables a continuous variable oening degree. The system has temerature and ressure sensors on each side of the comonents in the refrigeration cycle. Mass flow meters measures the mass flow rates of refrigerant in the refrigeration cycle and water on the secondary side of the evaorator. Temerature sensors measure the inlet and outlet temerature of the secondary media on resectively the evaorator and the condenser. The alied ower to the condenser fan and the comressor is measured. Finally the entire test system is located in a climate controlled room, such that the ambient temerature can be regulated. For data acquisition and control the XPC toolbox for SIMULINK is used. A. Model overview III. MODELING AND VERIFICATION A detailed model for an evaorator based on the conservation equations of mass, momentum and energy on the refrigerant, air and tube wall. This leads to a numerical solution of a set of differential equations discretized into a finite difference form, see [5]. This model gives very detailed information to the control designer comarable to the real system. This means that it is useful for testing of controllers, but due to the high comlexity not for design of new control rinciles. A simler model may be obtained by using a so called moving boundary model for the time deendent two hase flows and by assuming that satial variations in ressure are negligible, wich means that the momentum equation is no longer necessary. The numerical solution may describe the system quite well and results are shown in [2] and [3]. The moving boundary model is very general and may be fitted to most evaorator tyes. By simlifing of the moving boundary model further a very simle nonlinear model describing the dominating time constant and the nonlinear behavior between inut and outut is obtained. The gain and time constant variations as a function of the inuts and disturbances are exressed analytically. Following aroximations made are fluid flow is one-dimensional satial variations in ressure are negligible axial conduction is negligible cross sectional area of flow stream is constant the heat transfer coefficient from water to wall is small comared to the heat transfer coefficient from wall to boiling refrigerant the energy for suer heating the gas is negligible comared to the energy for evaorating the refrigerant the heat caacity of the wall between water and refrigerant is considered to be negligible. m& e hi Fig. 2. Two-hase section Te Le l e h g Suerheat section m& e ho Schematic drawing of the evaorator B. Energy balance two hase section dm e(h(t e,) ) = (h g h i )ṁ e α 1 Bl e (T water T e ) (1) The first term on the right side corresonds to the energy difference between the refrigerant leaving and entering the two hase section of the evaorator. The second term is the rate of the heat transfer from water to refrigerant. The left side describes the change of energy of the two hase section. M is the mass of refrigerant and h(t,v) is the enthaly which is a function of temerature T and secific volume v. From refrigerant data [1] we have h g = HDewP( ) h i = HBubP(P c ) h l = HBubP( ) T e = TDewP( ) v g = V DewP( ) v l = V BubP( ) From these values we have the quality of the liquid vaor system entering the two hase section (2) X i = h i h l h g h l (3) The secific volume of refrigerant entering the two hase region then becomes v i = X i v g + (1 X i )v l (4) The secific volume of refrigerant leaving the two hase region is v g. If the rate of evaoration is constant then the mean value of the secific volume becomes = (v i + v g )/2 = ((1 + X i )v g + (1 X i )v l )/2 (5) This gives the mass of refrigerant M e = BHl e (6) The left hand side of equation may now be calculated by using the chain rule for differentiation dm e(h(t e,) ) = (h(t e, ) ) dme d(h(t + M e,) ) e (7)

3 The assumtion of the same ressure in the evaorator means that the work associated with the time rate of change of ressure and internal energy of the accumulated refrigerant is negligible. This means that the equation becomes dm eh(t e,) = h(t e, ) dme (8) Calculations during the exeriments have shown that this aroximation is valid, see also [3]. Equation (1) then becomes h(t e, ) BH C. Mass balance dl e = (h g h i )ṁ e α 1 Bl e (T water T e ) (9) The mass balance for the evaorator is given by From dm e = ṁ e ṁ com (1) M e = V l v l + V g v g (11) where V l and V g are the volumen of liquid and gass resectively, (1) becomes d V l v l + d V g v g d = ṁ e ṁ com (12) d d Because V g V l and (12) becomes with κ = d 1 vg d. D. Suerheat section d Vg v g BHl e κ d d Vl v l (13) d d = ṁ e ṁ com (14) If the axial conduction is negligible and the heat caacity of the water c ṁ water >> c,e ṁ e the suerheat T SH becomes [ { }] T SH = (T water T e ) 1 ex α1b(le le)) c,eṁ e (15) E. Comressor The iston comressor model is develoed from factory given data as ṁ com = α c f c (16) where α c is a function of and P c. Assuming P c = P c,ref due to control of the condenser fan the variation of α c is only caused by variation of. In the working area for the system this variation is less than 5% and α c is considered as a constant. Equ. (16) in (12) then gives BHl e κ d α c f = + ṁe α cf c (17) c F. Combined model }] T e = TDewP( ) c 1 ẋ e = (h g h i )ṁ e c (T w T e )x e f c min 2 f c = + ṁe α cf[ c { T SH = (T w T e ) 1 ex 1 xe x δ with: a) c 1 = h(t e, ) BHL e b) c 2 = BHl e κ/(α c f min ) b) c = α 1 BL e c) x δ = c,e ṁ e /(α 1 BL e ) d) x e = l e /L e G. Control inut and measurement (18) The control inuts are f c,ref and ṁ e and the measured values are T SH, and T w. From these measurements the relative length x e of the two hase section is obtained by H. Model verification x e,meass = 1 x δ log T w T e T w T e T SH (19) The model arameters to be estimated are (BHL e,c 2 ) and θ = (α c,x δ,c ). A series of exeriments giving large signal excitation of the system for different working conditions are erformed. Simulation using the model equ. (18) with the same inut (ṁ ν,f c,ref ) as used in the exeriment then gives the outut (,T SH ). The time constants c 1 = H(T e,v e ) BHL e V e and c 2 are first found by visual fitting of simulated and measured values of the outut. Using these values for all exeriments θ may now be determined by minimizing the erformance function J(θ) = 1 t2 t 2 t 1 t 1 {K (,meass ) 2 + (T SH T SH,meass ) 2 } (2) where K = 5 gives a reasonable weigt between variation of and T SH. The result is shown in table I Simulated and measured values for exeriment 2 and 4 are shown in fig. (3) and (4). It is seen that the model gives a good descrition of the dominating dynamics of the system when otimized values are used. IV. NEW CONTROL METHODS The steady state value of the ressure given by the model c 2 f min f c P e = + ṁe α cf c,ref (21) is roortional to ṁ e /α c. In the model verification section the uncertenty of α c was shown. The refrigerant flow ṁ e was measured, but in a ractical control scheme an estimate of ṁ e has to be used. This means that the gain ṁ e /α c may have an error u to 3% of the best guess. Because the measured ressure is of good quality a way to overcome this roblem is to control the ressure by an PI-controller. The controller ṁ e = αcf c,ref τ 1+c 2 f min fc (,ref ) (22)

4 TABLE I EXPERIMENTS FOR MODEL VERIFICATION Exeriment BHL e c 2 α c x δ c J 1. f com,ref = 4 and.2 < ṁ ν < e f com,ref = 5 and.26 < ṁ ν < e f com,ref = 6 and.29 < ṁ ν < e ṁ ν =.22 and 35 < f com,ref < e ṁ ν =.28 and 45 < f com,ref < e random e random e random e mean values e [kg/s] : modelled and measured [bar] m ν T sh : modelled and measured 2 1 P e, ref f c - PI Equ.(22) m& e Model Equ.(18) Fig. 5. Model with PI control of inut m e. where the linear aroximation is valid over a wide range. The resulting cascaded structure shown in fig. (5) thin gives the following model for the relative filling x e and the evaoration temerature T SH Fig. 3. [bar] Fig Modeled and measured and T sh for variation of inut m ν f com,ref P : modelled and measured e T sh : modelled and measured Modeled and measured and T sh for variation of inut f com,ref gives the closed loo for the ressure τ = +,ref (23) The variation in the gain ṁ e /α 1 then only influence the time constant τ. The evaoration temerature T e based on the ressure may be calculated by T e = TDewP( ) a + a 1 (24) c 1 ẋ e = (h g h i )α c f c,ref c x e (T w + a a 1 ) τ P e = +,ref (25) In equation (25) x e has to be controlled to a value x e by,ref. If was the control inut it should be given the value Pe calculated by equ. (26) (h g h i )α c f c,ref P e c x e (T w +a a 1 P e ) = k 1 (x e x e) (26) This gives for constant x e c 1 (x e x e) = k 1 (x e x e) +((h g h i )α c f c,ref + c x e a 1 ) ( P e ) τ ( P e ) = ( P e ) +,ref P e τ P e (27) The ositive definite Lyaunov function candidate P = 1 2 c 1(x e x e) τ k 2 ( P e ) 2 (28) then has the time derivative P = k 1 (x e x e) 2 k 2 ( Pe ) 2 +( Pe )k 2 U U =,ref (1 + τ )Pe + (hg hi)αcf c,ref+c x ea 1 k 2 (x e x e) For a control inut P ref (29) P ref = (1 + τ )P e (hg hi)αcf c,ref+c x ea 1 k 2 (x e x e) (3) giving U = the time derivative of the Lyaunov function becomes P = k 1 (x e x e) 2 k 2 ( P e ) 2 (31)

5 Comressor seed: f c Comressor seed: f c Relative filling: x e Suerheat: T sh [Watt] Cooling caacity: Q e [W/s] Cooling caacity: Q e Fig. 6. simulated x e and for variation of inut m ν using the backsteing controller for known arameters Fig. 8. Measured f c, T sh and Q e using the backsteing controller Comressor seed: f c 6 3 Heat transfer coefficient: c 4 2 [watt/c] Relative filling: xe Cooling caacity: Q e 6 [W/s] Suerheat: T sh Cooling caacity: Q e [Watt] Fig. 9. Measured f c, T sh and Q e using a conventional TXV controller Fig. 7. simulated x e and for variation of c using the backsteing controller for constant c equal to the value before the change. This function is negative definite for ositive k 1 > and k 2 >, leading to a stable closed loo system. The new backsteing controller Pe = cxe(tw+a) k1(xe x e ) (h g h i)α cf c,ref +c x ea 1,ref = (1 + τ )Pe (hg hi)αcf c,ref+c x ea 1 = αcf f c,ref 1+c min 2 fc τ ṁ e (,ref ) ṁ e = sat(ṁ e,ṁ e min,ṁ e max ) k 2 (x e x e) (32) The develoed backsteing controller equ. (32) is tested on a simulation model based on estimated mean value model arameters. The result is shown in fig. 6 for the following controller arameters τ = 2 k 1 = 1 6 k 2 = 1 6 x e =.96 a = 26 a 1 = 8.5 (33) based on model knowledge. The variation in x e caused by the variation in m e is small due to the small time constant τ for the ressure controller. In the controller c is assumed known leading to a steady state m e equal to the reference. Fig. 7 shows the simulated outut if c is changed during the simulation. The figure shows no the need for an adatation of the c value. V. EXPERIMENTS Fig. 8 shows the erformance of the new controller for a ste change of the comressor seed f c between maximum and minimum. Only a small variation of the suerheat temerature

6 Suerheat: T sh Fig. 1. Measured T sh at startu. The lower curve is the new controller and the uer curve is the conventional TXV controller. [3] He, X.D., H.H. Asada, S. Liu and H. Itoh (1998a). Multivariable control of vaor comression systems. HVAC&R Research 4, [4] He, Xiang-Dong, Sheng Liu, Harry H. Assada and Hiroyuki Itoh (1998b). Multivariable control of vaor comression system. VAC&R Research. [5] Jia, X., C.P. Tso, P. Jolly and Y.W. Wong (1999). Distributed steady and dynamic modeling of dry-exansion evaorators. Int. Journal of Refrigeration 22, [6] Lei, Zhao and M. Zaheeruddin (25). Dynamic simulation and analysis of water chiller refrigeration system. Alied Thermal Engineering 25, [7] Parkum, J. and C. Wagner (1994). Identification and control of a dryexansion evaorator. 1th IFAC symosium on system identification. [8] Larsen, L.F.S (25). Model Based Control of Refrigeration Systems. Ph.D. Thesis ISBN Aalborg University / Danfoss A/S. [9] Larsen, L.F.S. and C. Thybo (24). Potential energy savings in refrigeration systems using otimal setoints. Conference on Control Alications, Taiei, Taiwan. [1] Skovru M. J. Thermodynamic and Thermohysical Proerties of Refrigerants. Ver. 3., 2, Technical University of Denmark. [11] Changquin Tian, Chuneng Dou, Xinjiang Yang and Xianting Li. Instability of automotive air conditioning system with a variable dislacement comressor. Part 1. Exerimental investigation, International Journal of Refrigeration, no28, 25. is seen even if the suerheat reference is as low as 5 degree Celsius. The figure also shows the variation of the cooling caacity. Fig. 9 shows the same exeriment for a conventional TXV controller. The disturbance of the suerheat temerature due to ste in comressor seed is significant comared to fig. 8 and may lead to hunting effects. Fig. 1 shows the startu of both the new controller and the TXV controller. The lower curve in the figure is the new controller and it is seen to obtain the steady state value faster than the TXV controller. This means that ulse wih modulation during low load may be more energy efficient with the new controller. VI. CONCLUSION A new control strategy is comared to a conventional control strategy based on a thermostatic exansion valve for control of the suerheat. A low order model for the highly nonlinear system with comressor seed and refrigerant flow as inuts and suerheat as outut is derived and verified exerimentally. The model has a form where backsteing may be used as a nonlinear design method. The develoed method gives a suerheat control which is nearly indeendent of the cooling caacity. The stability of the roosed method is validated theoretically by Lyaunov analysis and exerimental results shows the erformance of the system for a wide range of oerating oints. Comared to other methods no gain scheduling of the suerheat controller is necessary to cover a large region of oeration. The comarison between this new controller and the conventional TXV controller shows that continuous control is ossible for all values of the cooling caacity with the new controller. REFERENCES [1] Aaström, K.J. and B. Wittenmark. Adative Control - Second Edition, Addison-Wesley, [2] Grald, E.W. and J.W. MacArthur (1992). Moving boundary formulation for modeling time-deendent two-hase flows. Int. J. heat and Fluid Flow 13,