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Sub-optimal on–off switching control strategies for chilled water cooling systems with storage
PII:
Applied Thermal Engineering Vol. 18, No. 6, pp. 369±386, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-4311/98 $19.00 + 0.00 S1359-4311(97)00056-2
SUB-OPTIMAL ON±OFF SWITCHING CONTROL STRATEGIES FOR CHILLED WATER COOLING SYSTEMS WITH STORAGE Wei-Ling Jian and M. Zaheeruddin* Centre for Building Studies, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec, Canada, H3G 1M8 (Received 27 June 1997) AbstractÐA dynamic model of vapour compression refrigeration system is developed. The overall model consists of the following basic components: a compressor, a condenser, an expansion valve, an evaporator, an evaporative cooler and a cool storage. The integrated system is referred to as chilled water cooling system with storage (CWCS). The mathematical modelling of the CWC system undertaken in this study predicts the change in state of refrigerant in the system with respect to time. A computer program is developed to solve the dynamic equations along with empirical correlations describing refrigerant properties. Open-loop tests are carried out to study the performance characteristics of the system under varied cooling load and compressor speed. The model is intended to serve as an analytical design tool and to provide a basis for control analysis. Based on a heuristic method, `sub-optimal on± o control' strategies for the chilled water cool storage system are developed using a reduced order model. The methodology of generating such control pro®les is illustrated and the tests for optimality show that the control pro®les are near optimal. The on±o control scheme is simulated on the full order CWC system. The operating performance of the system is described under several simulated cases. The results show that the control scheme is capable of maintaining the chilled water temperature in the chosen range. # 1998 Elsevier Science Ltd. KeywordsÐSub-optimal control, on±o control, chilled water cooling systems, energy storage.
NOMENCLATURE A C COP Ctxv dT dt F ft h H1 J K Le M N P T t Ui Uimax Um Ummax Vc Vcl Vd Wc
heat transfer area of heat exchangers (m2) thermal capacity (kJ/K), speci®c heat (kJ kgÿ1 K), constant coecient of performance general ori®ce ¯ow area coecient temperature dierential time dierential p diameter of inner or outer wall of the tube (m) terminal time heat transfer coecient (kJ sÿ1 m2 K), enthalpy of refrigerant (kJ kgÿ1) enthalpy of refrigerant leaving the discharge port (kJ kg) energy cost, mechanical equivalent of heat polytrophic constant, assumed equal to speci®c heat ratio length of the evaporator tubes in evaporation region (m) mass ¯ow rate of refrigerant (kg hÿ1) compressor motor speed (rpm) pressure (Pa) temperature (8C), (K) time (s), (h) mass ¯ow rate ratio maximum mass ¯ow rate (kg hÿ1) mass ¯ow rate ratio of water from cooling coil maximum mass ¯ow rate of water from cooling coil (kg hÿ1) volume of condenser (m3) clearance volume of the compressor cylinder maximum volume of the compressor cylinder (m3) work done by compressor motor (kJ kgÿ1)
*To whom correspondence should be addressed. 369
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Subscripts ao ac c ci ch city chset chss co d e ei eo es g ic ie l ld max o oc oe ope pa r rc re s sh v w wa wac wae was wat wo A
outgoing air from evaporative cooler ambient air of the evaporative cooler condenser, condenser tubes, condensation, evaporative cooler inlet to the condenser chilled water city water setpoint value of chilled water temperature steady-state condition of chilled water outlet of the condenser discharge section evaporator tubes in evaporation region, evaporation inlet of the evaporator outlet of evaporator evaporator tubes in superheat region refrigerant vapour inside of condenser tubes inside of evaporator tubes refrigerant liquid lead maximum outlet between the condenser tubes and water between the evaporator tubes and water operation condition air cross section of inner tube of evaporator refrigerant in condenser refrigerant in evaporator suction, superheat superheat saturated refrigerant vapour condenser and evaporator tube wall water cycling water in condenser cycling water in evaporator cycling water in superheat region of evaporator water water in evaporative cooler environment
Greek a D Zv x r t F
Mean void fraction, weighting factors Dierence clearance volumetric eciency heat exchanger eectiveness density (kg mÿ3) time constant mass ¯ow rate of refrigerant (kg hÿ1)
1. INTRODUCTION
Optimal control as a means of improving the energy eciency of Heating, Ventilating and Air conditioning (HVAC) systems has been demonstrated in recent studies (Zaheeruddin et al., 1990; Zaheeruddin and Wang, 1992). However, ®nding optimal control solutions to nonlinear large scale HVAC systems is a dicult and computationally extensive search problem. Furthermore, on-line implementation poses severe limitations on the use of optimal control strategies and requires expensive hardware for successful implementation. In small to medium sized HVAC applications, there is a need to develop cost eective, reliable and yet near optimal control strategies. It is with this objective in mind that we propose a sub-optimal on±o switching control technique for HVAC applications. To focus on the application of the proposed method, a chilled water cooling (CWCS) system with storage (Fig. 1) is considered. The main elements of the system are (1) a reciprocating compressor, (2) a shell-and-tube condenser, (3) a dry expansion evaporator, (4) a thermostatic expansion valve, (5) an evaporative cooler, and (6) a chilled water storage tank. The vapour
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compression refrigeration system comprises of elements (1) to (4). There are four controllers in the system: U1 for evaporative cooler fan, U2 for circulation water loop between condenser and evaporative cooler, U3 for the energy input to the compressor and U4 for chilled water loop between evaporator and storage tank. Although four controllers are needed for individual local control actions, the most dominant from the energy viewpoint is the power input to the compressor (U3). Therefore the optimization problem posed in this study focuses on the minimization of U3, while holding U1, U2 and U4 at their nominal values. In response to the anticipated cooling load on the building, it is of interest to charge the storage tank ahead of time so that peak electrical demand on the building is reduced and the cooling load is met in the most energy-ecient manner. To address this issue, a dynamic model of CWC system, shown in Fig. 1, is developed (Section 2). The statement of the control problem, its formulation and solution technique and the results obtained are discussed in Section 3, followed by implementation issues (Section 4) and conclusions in Section 5.
Fig. 1. Schematic diagram of the CWC system with storage.
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2. DYNAMIC MODEL OF A CWC SYSTEM WITH STORAGE
Several researchers have developed transient models of vapour compression refrigeration system (Dhar, 1978; Chi and Didion, 1982; Yasuda and Touber, 1982; Jian, 1996) with varying degree of complexity. To model a CWC system, it is necessary to consider not only the vapour compression system but also an evaporative cooler, a storage tank and building load. To this end, a dynamic model of the overall CWC system, which is useful for control analysis, was developed. In the following, the equations describing the component models are given.
2.1. Dynamic model of a CWC system with storage The clearance volumetric eciency, Zv, is expressed as " # Vcl pd 1=K ÿ1 Zv 1 ÿ Vd ps
1
Mass ¯ow rate of refrigerant through the compressor, fv (kg hÿ1), is written as fv rs Vd NZv
2
ÿ1
The work done, Wc (kJ kg ), of compressor motor is given by " Kÿ1 # K pd K Wc ÿ1 Ps Vs ps K ÿ1
3
The enthalpy of the superheat vapour, Hl (kJ kgÿ1), leaving the discharge port is given by H l H s Wc The temperature, T1 of vapour discharged by the compressor is given by Kÿ1 pd K Tl Ts ps
4
5
2.2. Condenser model The energy balance equation for refrigerant is dr drl dTrc ac rv Cv 1 ÿ ac rl Cl ac hv v 1 ÿ ac hl dTrc dTrc dt 1 fi hci ÿ fo hco ÿ hic Aic Trc ÿ Tc Vc
6
The energy balance equation at the tube surface is written as Cc
dTc hic Aic Trc ÿ Tc ÿ hoc Aoc Tc ÿ Twac dt
7
The energy balance equation for cooling water is given by Cwa
dTwac hoc Aoc Tc ÿ Twac ÿ U2 U2max Cwat Twac ÿ Two dt
8
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2.3. Evaporator model 2.3.1. Evaporation region. The energy balance equation for the refrigerant is given in a combined form as drl drl dTre hv 1 ÿ ae hl Ar Le ae rv Cv 1 ÿ ae rl Cl ae dTre dTre dt fl hei ÿ fv heo he Aie Te ÿ Tre The energy balance equation at the tube wall is dTe hwa Foe Twae ÿ Te ÿ he Fie Te ÿ Tre rw Cw Aw dt
9
10
Chilled water in evaporation region is considered as a individual control volume. The corresponding energy balance equation is dTwae U4 U4max Cwat Tch ÿ Twae ÿ hwa Foe Twae ÿ Te 11 rwa Cwa Awa dt Le 2.3.2. Superheat region. Inside this region, the superheated vapour refrigerant is assumed incompressible. The energy balance equation for superheated vapour is @Ts t; y @Ts t; y 12 fv Cs hes Fie Tes t; y ÿ Ts t; y rs Cs Ar @t @y The energy balance equation at the tube wall is @Tes t; y hwa Foe Twas t; y ÿ Tes t; y ÿ hes Fie Tes t; y ÿ Ts t; y rw Cw Aw @t
13
The energy balance on chilled water is @Twas t; y U4 U4max Twae ÿ Twas t; y rwa Awa Dy @t ÿ
hwa Foe Twas t; y ÿ Tes t; y rwa Cwat Awa
14
2.4. Thermostatic expansion valve model The general equation of TXV is given by r;txv Ctxv DTope ÿ DTss r DPtxv 1=2 M l
15
DTope Tsh ÿ Tre
16
where The available pressure drop across the TXV valve DPtxv is computed from DPtxv Pd ÿ Ps
17
2.5. Evaporative cooler model The energy balance equation for the circulating water between condenser and evaporative cooler is
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dTwo Tao Tac U2 U2max Cwat x Tc ÿ Two ÿ Hc Ac Two ÿ Cwo dt 2
18
where Tao Tac
U2 U2max Cwat x Te ÿ Two U1 U1max Cpa
19
2.6. Chilled water storage tank model If Tch is the chilled water temperature and Cch is the thermal capacity of the storage tank, the energy balance on the tank is Cch
dTch ÿU4 U4max Cwat Tch ÿ Twas i Um Ummax Cwat Tcity ÿ Tch dt
20
2.7. Open-loop responses of the CWC system The dynamic equations of CWC system were discretized in space and time. A total of 49 equations were solved at each time step using the Gear method of numerical integration (Gear, 1971). A 4.5 ton (15.8 kW) CWC system with parameters listed in Table 1 was simulated. Several open-loop responses (responses to constant inputs) were examined to study the dynamic characteristics of the system. A few results are shown here. Figure 2 shows refrigerant temperatures at four critical points within the refrigerant circuitry. Among four curves, the one with highest value is the compressor outlet temperature. The refrigerant leaving the compressor as a gas reaches a temperature of 558C. The condenser outlet temperature, which is that of liquid refrigerant leaving the condenser, reaches a steady-state condition of about 408C. The other two curves show the steady-state temperatures at the compressor inlet and evaporator inlet. Both of them drop o gradually from the initial value, and reach the steady-state value of 68C and ÿ38C respectively. After the entire amount of refrigerant liquid is evaporated, the temperature starts rising until 98C, which is superheat temperature of vapour entering the compressor. The eect of compressor motor speed N on chilled water temperature Tch is plotted for three loads (100%, 80%, and 60% of full load) in Fig. 3. It is obvious that as load decreases and motor speed increases, the chilled water temperature is decreased. If load is constant, increasing compressor motor speed would generate a higher quantity of gas refrigerant ¯ow rate, therefore increasing the refrigerating capacity of the system. In other words, more cool energy is produced which results in lower chilled water temperature. On the other hand, if the compressor motor
Table 1. List of parameters Symbol
Magnitude
Units
ANmax Vcl Vd Cch U2max U3max U4max Ummax Aic Aoc Aie Aoe COPmax Tchset
1200.0 0.05 0.0006 8000.0 7000.0 30,000 6000.0 1150.0 15.0 2.33 0.47 1.06 3 8
rpm Dimensionless m3 kJ8Cÿ1 kg hÿ1 kJ hÿ1 kg hÿ1 kg hÿ1 m2 m2 m2 m2 Dimensionless 8C
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Fig. 2. Refrigerant states as function of time.
speed is held constant, but more cooling is required, the system has to generate higher cooling capacity in order to extract more heat through the evaporator, resulting in higher evaporation temperature and, hence, higher temperature of chilled water.
Fig. 3. Tch as function of motor speed and percent load.
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The open-loop simulation results presented above give the transient response characteristics of the system such as steady-state time. The developed model is useful in assessing the range of design parameters under various load conditions.
3. SUB-OPTIMAL ON±OFF CONTROL STRATEGIES
For the system dynamics given by Equations 1)±(20, an optimal control problem can be formulated and solved numerically to determine optimal control trajectories. Such methods are widely used (Braun et al., 1989). Diculties arise with regard to computational time, convergence and concerns about global optimality of the solution. These concerns and hardware costs limit their application on real systems. To this end, the objective of this study is to develop optimal or sub-optimal on±o control strategies. In order to develop sub-optimal on±o control strategies a two-tier scheme is proposed. First, a sub-optimal on±o control strategy based on a reduced-order CWC system is designed. The strategy so designed is then used to simulate the full order CWC system described by Equations 1)±(20 and the results are compared. In the following section, the reduced order model equations are developed, the optimal on±o control problem is formulated and a solution technique is described. 3.1. Construction of reduced-order model If Tch is the chilled water temperature and Cch is the storage capacity, the energy balance equation on the storage tank (Fig. 1) is written as Cch
dTch ÿU3 U3max COP Um Ummax Cwat T1 ÿ Tch dt
21
where the rate of energy stored in the storage tank is equal to the rate of energy extracted from the chilled water U3U3max COP and the heat gains from the return water UmUmmaxCwat(TAÿTch). Um in the equation is the normalized mass ¯ow rate of chilled water supplied to cooling coil, and U3 is the normalized input energy to the chiller. The coecient of performance (COP) of the chiller is modelled as (Jian, 1996). T1 ÿ Tch 22 COP COPmax ÿ 1 1 ÿ DTmax where TA is the sink temperature and D Tmax is the maximum temperature dierential the chiller is designed to work with. The cost function for minimizing the energy use is de®ned as Z tz a1 U3 dt a2 Tch ÿ Tchset 2 dt 23 J t0
The purpose of this function is in evaluating and minimizing the energy cost of operating the chiller (U3) and satisfying the setpoint temperature. The term a1U3 dt represents the amount of chiller energy input, a2(TchÿTchset)2 dt is used to minimize the temperature variations; a1 and a2 are weighting factors. Assuming the cooling load is known a priori, we propose a periodic solution for this problem to meet the following condition Tch tf ÿ Tchss 20:05 C
24
where Tchss is steady-state temperature. The constraint on the input capacity of the chiller U3 is taken to be 0 U3 U3max
25
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Fig. 4. Optimal strategy for load pro®le1. (a) Load pro®le1, (b) Tch response, (c) U3 pro®le, (d) cost function.
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3.2. Heuristic on±o control technique Consider a load pro®le as shown in Fig. 4. Since the load is time varying as shown in Fig. 4(a), we have to look for dynamic optimal solution for it. Based on heuristic ideas, we present a set of `optimal' switching modes consisting of fully on portion and unsaturated steadystate pro®les. The objective of the optimal switching strategy is that is should satisfy the boundary condition (24) and the minimum cost criterion. Figure 4(c) shows the optimal switching control pro®le for operating the chiller U3 (in solid line) corresponding to the load pro®le shown in Fig. 4(a). The method used to generate this control pro®le consists of the following two steps: (1) ®nding the steady-state optimal solution and (2) determining the lead time tld and on time ton of the system such that (a) the boundary condition is satis®ed and (b) the minimum cost criterion is met. 3.3. ON time search methodology A numerical search technique has to be used in order to determine the lead time and on time shown in Fig. 4(c). In the methodology used, the search problem is considered as a one dimensional search problem. In Fig. 4(a), since the load is increasing at 1.5 h, the chilled water storage has to be charged ahead of time so that water is available for supply at the set point temperature right at 1.5 h. With this assumption, U3 is turned on at time tld, and it is held at maximum value for period of `on' time ton, so that the chilled water temperature returns to its steady-state value at exactly 2.5 h. With tld and ton thus determined, the total cost was computed. 3.4. Test for optimality of control pro®les It is reasonable to ask whether the control pro®le shown in Fig. 4(c) (solid line) is optimal because there could be other solutions with dierent lead time tld. In order to answer this question, an extended search method was used. This time, we again use the numerical search method, but the problem is extended to a two variable search problem. The search for both t*ld and t*on was executed over a wide range, that is, from one hour before and half hour later from the point of application of load (1.5 h in Fig. 4a). For each time interval dt2, a search for the on time that minimized the cost was made. When starting the search procedure, an initial value for lead time was assumed and U3 is turned on, then for each small interval dt2, a search for ton was made such that the state variable returns to its steady-state value at 2.5 h, at this point U3 is set back to its steady-state value. If such ton is determined, one set of iteration has been completed for the lead time that was set at the beginning, and the corresponding cost value J is calculated for that iteration. The next round of searching will move ahead to the new lead time by adding a small amount dt1 to the initial value, and this process will continue until the lead time is equal to 2.0 h. While each step of searching is executed, the cost J was found. The result for optimality of the control method is shown in Fig. 4(d). In this ®gure, the cost J is plotted as a function of both lead time and on time. As it is shown in the ®gure, for each individual tld, there is a ton which causes Tch to go to steady-state value at 2.5 h. However, there is only one set of tld and ton that gives minimum cost Jmin (4.06). The switching control pro®le for U3 and the temperature Tch response for this optimal case are shown in Fig. 4(b, c) (solid line). 3.5. Further investigation of the on±o control pro®les In order to test the methodology of generating the heuristic on±o control pro®les for dierent load conditions, several runs were made by changing the load con®gurations. Since the ®rst load pro®le shown in Fig. 4(a) was referred to as case1, the new load is identi®ed here as case2 load pro®le as shown in Fig. 5(a). Although this load pro®le is somewhat dierent than the ®rst one, the same methodology for computing optimal on±o control solution will be used. For load pro®le shown in Fig. 5(a), again we expect to obtain the heuristic on±o control strategy for U3, which satis®es the boundary condition (24) and meets the minimum cost criterion expressed in (23). Figure 5(b) shows the cost J as function of tld and ton for case2. Results show that for lead time ranging between ÿ1.4 and 0.4 h, a convex function is obtained, and there is a point at
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Fig. 5. Chiller energy cost as function of lead time and on time for load pro®le2. (a) Load pro®le2, (b) cost function.
which the minimum cost Jmin accrues. The test results shown above demonstrate that the technique based on one-dimensional two variable search gave minimum cost under ¯uctuating load conditions. 3.6. Application of heuristic on±o control pro®les It is of interest to extend the heuristic on±o control technique described in the last section for a typical day operation. To this end, the ®rst step that is necessary is to determine the optimal switching solution over intervals of 1±2 h until a 24 h period is covered. 3.6.1. Multiple usage of heuristic on±o control technique for daily load. Consider a step change load pro®le as shown in Fig. 6(a). For the day with this load curve, known a priori, we wish to ®nd the heuristic switching control sequence for U3. In this case, the cost function for minimizing the 24 h pattern of energy use is de®ned as
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Fig. 6. Optimal unconstrained Tch and U3 responses. (a) Daily load case1 pro®le, (b) Tch response, (c) U3 response.
Z J
24 0
a1 U3 dt Tch ÿ Tchset 2 dt
26
The terms in Equation 26 are the same as those in Equation 23. We also expect the solution to this problem to satisfy the following boundary condition Tch 22 ÿ Tchss 20:05 C Moreover, the constraint on the chilled water temperature Tch is taken to be
27
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7:0 C Tch 9:0 C
381
28
For the daily load pro®le shown in Fig. 6(a), we propose an optimal unconstrained control solution by repeated application of the heuristic on±o control technique, more speci®cally, once every 2 h, between 10 and 22 h. In this way, the control solution for U3 which satis®es boundary condition (27) and meets the minimum cost criterion (26) was found. Meanwhile, the minimum cost Jmin for the day would be computed by integration of minimum cost values of each searched interval. Figure 6(b, c) presents the on±o control pro®le of U3 and resulting chilled water temperature Tch. As we see in the Fig. 6(b), the temperature Tch is beyond the constraint
Fig. 7. Open loop responses of Tch for three load cases. (a) Tch response at Um=0.3, (b) Tch response at Um=0.5, (c) Tch response at Um=0.8.
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W.-L. Jian and M. Zaheeruddin
limit (28). However, the magnitude of net overshoot Tos is found to be relatively small (0.35± 0.588C), and these overshoots occur between 12 and 18 h, which indicates that during the high load period, direct application of the control technique developed earlier could lead to some overshoot. It is possible to search for a new lead time to reduce the overshoots. It was found that one or two trials were enough to get a constraint solution with minimal or no overshoot in Tch. 3.7. Simulation of switching control strategies on full-order system In the above we have presented a methodology by which optimal on±o control strategies for a chilled water system can be computed. However, to illustrate the technique we chose a reduced-order model. It is of interest to see if the optimal on±o control strategy determined from the reduced-order model can be successfully simulated on the full-order model developed in Section 2. Before doing this it would be necessary to make sure that the reduced-order model behaves in a similar way to the full-order model. To do this, ®rst we need to make sure that the
Fig. 8. Comparison of Tch responses between two models for load pro®le2. (a) Load pro®le2, (b) Tch response.
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parameters in the reduced-order model are properly chosen so that the open loop responses are identical or close to that of the full-order model. The resulting open loop responses of chilled water temperature Tch for both models are shown in Fig. 7(a±c), for three dierent loads Um=0.3, Um=0.5 and Um=0.8. It is apparent that steady-state time is the same for both models. However, the trajectories of Tch are somewhat dierent, and for larger load Um=0.8, the dierence between the two Tch curves is more evident. This is because the full-order model is large and complex as such the system characteristics are eected by several parameters compared to the reduced-order model. Therefore, it can be considered that both models are close enough for the purpose of designing optimal on±o control using the reduced-order model.
Fig. 9. Comparison of Tch and energy input response of two models for daily load (case1). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
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3.7.1. Results. Simulation results are presented for load pro®le shown in Fig. 8(a). First the optimal on±o control strategy was found by using the reduced-order model. Then this optimal on±o control sequence was used to simulate the full-order system response. The resulting Tch pro®les were compared. We can see from the ®gure that the temperature Tch from full-order model returns to the steady-state value exactly at 2.5 h, thus satisfying the boundary condition (24). In addition, it is found that when load remains low, say during period of 0±1.5 h and 2.5± 3.5 h, the Tch response of the full-order model is quite close to that of reduced-order model; however, when load increases, the temperature Tch of the full-order model is obviously lower than that of the reduced-order model. This could be explained in that the full-order model was
Fig. 10. Comparison of Tch and energy input response of two models for daily load (case2). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
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Fig. 11. Comparison of Tch and energy input reponses of two models for daily load (case3). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
developed more thoroughly and is considered to be a more accurate representation of system dynamics. The maximum temperature dierence in Tch between the two models is about 0.68C for load pro®le, shown in Fig. 8(a). The simulation results for the daily load are shown in Fig. 9Fig. 10Fig. 11(a±c)±11(a±c). It can be noted that by implementing the near-optimal on±o control pro®les on the full-order model, the response of Tch is basically kept close within the constraint. The comparison of temperature Tch pro®les between the two models is also given in the ®gures. They indicate that both Tch curves look very close and the largest dierence between them is 0.15±0.258C. These results show that the heuristic on±o control methodology developed in this study is simple and near optimal, and can be used for operation of chilled water cooling systems with storage.
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4. IMPLEMENTATION ISSUES
From the past history of load patterns and the anticipated load for the next day, the optimal lead time, switching times, and the resulting chilled water temperature pro®le can be computed ahead of time by using the methodology described in Section 3. In terms of implementation, it would be easier to track the chilled water temperature pro®le by using a two-position controller and a timer. The clock enables the controller to start the compressor at the computed lead time of the day. Thereafter, hourly optimal high and low limits of the chilled water temperature could be read from a reference table for eecting the on±o control sequence. Since the predicted load is likely to be dierent from the actual load, it may be necessary to ®ne tune the high and low limits somewhat, based on the actual load changes. 5. CONCLUSIONS
A dynamic model of a chilled water cooling system with storage useful for design and control analysis has been developed. The development of a sub-optimal on±o control strategy based on a reduced-order model and the simulation of on±o control strategies on the full-order chilled water cooling system have been studied. The results show that: (1) the sub-optimal on± o control strategy developed based on one dimensional search method has shown to be `optimal' when the chiller system is operating under time varying load, (2) the daily chilled water temperature response shows that the multiple application of the unconstrained optimal on±o control strategy with some ®ne tuning can give good control of chilled water temperature, (3) the results of simulation of the sub-optimal on±o control pro®le on the full order model show that, for the short period load con®gurations, the maximum dierence in the chilled water temperature between full and reduced order model are 0.15, 0.5, and 0.38C; for the daily load pro®les the dierences are 0.2, 0.18, and 0.18C. Being simple and sub-optimal, the on±o control technique can be successfully applied to control chilled water cooling systems. AcknowledgementsÐThis work was supported by funds (OGP0036380) from the Natural Sciences and Engineering Research Council of Canada.
REFERENCES 1. J. E. Braun, S. A. Klein, W. A. Beckman and J. W. Mitchell, Methodologies for optimal control of chilled water systems without storage. ASHRAE Transactions 95(1), 652±662 (1989). 2. J. Chi and D. Didion, A simulation model of the transient performance of heat pumps. International Journal of Refrigeration 5(3), pp. 176±184 (1982). 3. Dhar, M., Transient analysis of refrigeration system, Ph.D. Thesis. Purdue University (1978). 4. Gear, C. W., Numerical initial value problems in ordinary dierential equations. Prentice-Hall (1971). 5. Jian, W. L., Dynamic modelling and on±o switching control of a chilled water cooling system with storage, Master's thesis. Centre for Building Studies, Concordia University, Montreal, Canada (1996). 6. Yasuda, H. and Touber, S., Simulation model of a vapour compression refrigeration system. Laboratory for Refrigerating Engineering and Air Conditioning Rept. No. WTHD 133. Department of Mechanical Engineering, Delft University of Technology, The Netherlands (1982). 7. M. Zaheer-uddin and J. C. Y. Wang, Start-stop control strategies for heat recovery in multi-zone water-loop heat pump systems. Heat Recovery Systems & CHP 12(4), (1992). 8. M. Zaheer-uddin, R. E. Rink and V. G. Gourishankar, Heuristic control pro®les for integrated boilers. ASHRAE Transactions 96(2), 205±211 (1990).
Applied Thermal Engineering Vol. 18, No. 6, pp. 369±386, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-4311/98 $19.00 + 0.00 S1359-4311(97)00056-2
SUB-OPTIMAL ON±OFF SWITCHING CONTROL STRATEGIES FOR CHILLED WATER COOLING SYSTEMS WITH STORAGE Wei-Ling Jian and M. Zaheeruddin* Centre for Building Studies, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec, Canada, H3G 1M8 (Received 27 June 1997) AbstractÐA dynamic model of vapour compression refrigeration system is developed. The overall model consists of the following basic components: a compressor, a condenser, an expansion valve, an evaporator, an evaporative cooler and a cool storage. The integrated system is referred to as chilled water cooling system with storage (CWCS). The mathematical modelling of the CWC system undertaken in this study predicts the change in state of refrigerant in the system with respect to time. A computer program is developed to solve the dynamic equations along with empirical correlations describing refrigerant properties. Open-loop tests are carried out to study the performance characteristics of the system under varied cooling load and compressor speed. The model is intended to serve as an analytical design tool and to provide a basis for control analysis. Based on a heuristic method, `sub-optimal on± o control' strategies for the chilled water cool storage system are developed using a reduced order model. The methodology of generating such control pro®les is illustrated and the tests for optimality show that the control pro®les are near optimal. The on±o control scheme is simulated on the full order CWC system. The operating performance of the system is described under several simulated cases. The results show that the control scheme is capable of maintaining the chilled water temperature in the chosen range. # 1998 Elsevier Science Ltd. KeywordsÐSub-optimal control, on±o control, chilled water cooling systems, energy storage.
NOMENCLATURE A C COP Ctxv dT dt F ft h H1 J K Le M N P T t Ui Uimax Um Ummax Vc Vcl Vd Wc
heat transfer area of heat exchangers (m2) thermal capacity (kJ/K), speci®c heat (kJ kgÿ1 K), constant coecient of performance general ori®ce ¯ow area coecient temperature dierential time dierential p diameter of inner or outer wall of the tube (m) terminal time heat transfer coecient (kJ sÿ1 m2 K), enthalpy of refrigerant (kJ kgÿ1) enthalpy of refrigerant leaving the discharge port (kJ kg) energy cost, mechanical equivalent of heat polytrophic constant, assumed equal to speci®c heat ratio length of the evaporator tubes in evaporation region (m) mass ¯ow rate of refrigerant (kg hÿ1) compressor motor speed (rpm) pressure (Pa) temperature (8C), (K) time (s), (h) mass ¯ow rate ratio maximum mass ¯ow rate (kg hÿ1) mass ¯ow rate ratio of water from cooling coil maximum mass ¯ow rate of water from cooling coil (kg hÿ1) volume of condenser (m3) clearance volume of the compressor cylinder maximum volume of the compressor cylinder (m3) work done by compressor motor (kJ kgÿ1)
*To whom correspondence should be addressed. 369
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Subscripts ao ac c ci ch city chset chss co d e ei eo es g ic ie l ld max o oc oe ope pa r rc re s sh v w wa wac wae was wat wo A
outgoing air from evaporative cooler ambient air of the evaporative cooler condenser, condenser tubes, condensation, evaporative cooler inlet to the condenser chilled water city water setpoint value of chilled water temperature steady-state condition of chilled water outlet of the condenser discharge section evaporator tubes in evaporation region, evaporation inlet of the evaporator outlet of evaporator evaporator tubes in superheat region refrigerant vapour inside of condenser tubes inside of evaporator tubes refrigerant liquid lead maximum outlet between the condenser tubes and water between the evaporator tubes and water operation condition air cross section of inner tube of evaporator refrigerant in condenser refrigerant in evaporator suction, superheat superheat saturated refrigerant vapour condenser and evaporator tube wall water cycling water in condenser cycling water in evaporator cycling water in superheat region of evaporator water water in evaporative cooler environment
Greek a D Zv x r t F
Mean void fraction, weighting factors Dierence clearance volumetric eciency heat exchanger eectiveness density (kg mÿ3) time constant mass ¯ow rate of refrigerant (kg hÿ1)
1. INTRODUCTION
Optimal control as a means of improving the energy eciency of Heating, Ventilating and Air conditioning (HVAC) systems has been demonstrated in recent studies (Zaheeruddin et al., 1990; Zaheeruddin and Wang, 1992). However, ®nding optimal control solutions to nonlinear large scale HVAC systems is a dicult and computationally extensive search problem. Furthermore, on-line implementation poses severe limitations on the use of optimal control strategies and requires expensive hardware for successful implementation. In small to medium sized HVAC applications, there is a need to develop cost eective, reliable and yet near optimal control strategies. It is with this objective in mind that we propose a sub-optimal on±o switching control technique for HVAC applications. To focus on the application of the proposed method, a chilled water cooling (CWCS) system with storage (Fig. 1) is considered. The main elements of the system are (1) a reciprocating compressor, (2) a shell-and-tube condenser, (3) a dry expansion evaporator, (4) a thermostatic expansion valve, (5) an evaporative cooler, and (6) a chilled water storage tank. The vapour
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compression refrigeration system comprises of elements (1) to (4). There are four controllers in the system: U1 for evaporative cooler fan, U2 for circulation water loop between condenser and evaporative cooler, U3 for the energy input to the compressor and U4 for chilled water loop between evaporator and storage tank. Although four controllers are needed for individual local control actions, the most dominant from the energy viewpoint is the power input to the compressor (U3). Therefore the optimization problem posed in this study focuses on the minimization of U3, while holding U1, U2 and U4 at their nominal values. In response to the anticipated cooling load on the building, it is of interest to charge the storage tank ahead of time so that peak electrical demand on the building is reduced and the cooling load is met in the most energy-ecient manner. To address this issue, a dynamic model of CWC system, shown in Fig. 1, is developed (Section 2). The statement of the control problem, its formulation and solution technique and the results obtained are discussed in Section 3, followed by implementation issues (Section 4) and conclusions in Section 5.
Fig. 1. Schematic diagram of the CWC system with storage.
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2. DYNAMIC MODEL OF A CWC SYSTEM WITH STORAGE
Several researchers have developed transient models of vapour compression refrigeration system (Dhar, 1978; Chi and Didion, 1982; Yasuda and Touber, 1982; Jian, 1996) with varying degree of complexity. To model a CWC system, it is necessary to consider not only the vapour compression system but also an evaporative cooler, a storage tank and building load. To this end, a dynamic model of the overall CWC system, which is useful for control analysis, was developed. In the following, the equations describing the component models are given.
2.1. Dynamic model of a CWC system with storage The clearance volumetric eciency, Zv, is expressed as " # Vcl pd 1=K ÿ1 Zv 1 ÿ Vd ps
1
Mass ¯ow rate of refrigerant through the compressor, fv (kg hÿ1), is written as fv rs Vd NZv
2
ÿ1
The work done, Wc (kJ kg ), of compressor motor is given by " Kÿ1 # K pd K Wc ÿ1 Ps Vs ps K ÿ1
3
The enthalpy of the superheat vapour, Hl (kJ kgÿ1), leaving the discharge port is given by H l H s Wc The temperature, T1 of vapour discharged by the compressor is given by Kÿ1 pd K Tl Ts ps
4
5
2.2. Condenser model The energy balance equation for refrigerant is dr drl dTrc ac rv Cv 1 ÿ ac rl Cl ac hv v 1 ÿ ac hl dTrc dTrc dt 1 fi hci ÿ fo hco ÿ hic Aic Trc ÿ Tc Vc
6
The energy balance equation at the tube surface is written as Cc
dTc hic Aic Trc ÿ Tc ÿ hoc Aoc Tc ÿ Twac dt
7
The energy balance equation for cooling water is given by Cwa
dTwac hoc Aoc Tc ÿ Twac ÿ U2 U2max Cwat Twac ÿ Two dt
8
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2.3. Evaporator model 2.3.1. Evaporation region. The energy balance equation for the refrigerant is given in a combined form as drl drl dTre hv 1 ÿ ae hl Ar Le ae rv Cv 1 ÿ ae rl Cl ae dTre dTre dt fl hei ÿ fv heo he Aie Te ÿ Tre The energy balance equation at the tube wall is dTe hwa Foe Twae ÿ Te ÿ he Fie Te ÿ Tre rw Cw Aw dt
9
10
Chilled water in evaporation region is considered as a individual control volume. The corresponding energy balance equation is dTwae U4 U4max Cwat Tch ÿ Twae ÿ hwa Foe Twae ÿ Te 11 rwa Cwa Awa dt Le 2.3.2. Superheat region. Inside this region, the superheated vapour refrigerant is assumed incompressible. The energy balance equation for superheated vapour is @Ts t; y @Ts t; y 12 fv Cs hes Fie Tes t; y ÿ Ts t; y rs Cs Ar @t @y The energy balance equation at the tube wall is @Tes t; y hwa Foe Twas t; y ÿ Tes t; y ÿ hes Fie Tes t; y ÿ Ts t; y rw Cw Aw @t
13
The energy balance on chilled water is @Twas t; y U4 U4max Twae ÿ Twas t; y rwa Awa Dy @t ÿ
hwa Foe Twas t; y ÿ Tes t; y rwa Cwat Awa
14
2.4. Thermostatic expansion valve model The general equation of TXV is given by r;txv Ctxv DTope ÿ DTss r DPtxv 1=2 M l
15
DTope Tsh ÿ Tre
16
where The available pressure drop across the TXV valve DPtxv is computed from DPtxv Pd ÿ Ps
17
2.5. Evaporative cooler model The energy balance equation for the circulating water between condenser and evaporative cooler is
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dTwo Tao Tac U2 U2max Cwat x Tc ÿ Two ÿ Hc Ac Two ÿ Cwo dt 2
18
where Tao Tac
U2 U2max Cwat x Te ÿ Two U1 U1max Cpa
19
2.6. Chilled water storage tank model If Tch is the chilled water temperature and Cch is the thermal capacity of the storage tank, the energy balance on the tank is Cch
dTch ÿU4 U4max Cwat Tch ÿ Twas i Um Ummax Cwat Tcity ÿ Tch dt
20
2.7. Open-loop responses of the CWC system The dynamic equations of CWC system were discretized in space and time. A total of 49 equations were solved at each time step using the Gear method of numerical integration (Gear, 1971). A 4.5 ton (15.8 kW) CWC system with parameters listed in Table 1 was simulated. Several open-loop responses (responses to constant inputs) were examined to study the dynamic characteristics of the system. A few results are shown here. Figure 2 shows refrigerant temperatures at four critical points within the refrigerant circuitry. Among four curves, the one with highest value is the compressor outlet temperature. The refrigerant leaving the compressor as a gas reaches a temperature of 558C. The condenser outlet temperature, which is that of liquid refrigerant leaving the condenser, reaches a steady-state condition of about 408C. The other two curves show the steady-state temperatures at the compressor inlet and evaporator inlet. Both of them drop o gradually from the initial value, and reach the steady-state value of 68C and ÿ38C respectively. After the entire amount of refrigerant liquid is evaporated, the temperature starts rising until 98C, which is superheat temperature of vapour entering the compressor. The eect of compressor motor speed N on chilled water temperature Tch is plotted for three loads (100%, 80%, and 60% of full load) in Fig. 3. It is obvious that as load decreases and motor speed increases, the chilled water temperature is decreased. If load is constant, increasing compressor motor speed would generate a higher quantity of gas refrigerant ¯ow rate, therefore increasing the refrigerating capacity of the system. In other words, more cool energy is produced which results in lower chilled water temperature. On the other hand, if the compressor motor
Table 1. List of parameters Symbol
Magnitude
Units
ANmax Vcl Vd Cch U2max U3max U4max Ummax Aic Aoc Aie Aoe COPmax Tchset
1200.0 0.05 0.0006 8000.0 7000.0 30,000 6000.0 1150.0 15.0 2.33 0.47 1.06 3 8
rpm Dimensionless m3 kJ8Cÿ1 kg hÿ1 kJ hÿ1 kg hÿ1 kg hÿ1 m2 m2 m2 m2 Dimensionless 8C
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Fig. 2. Refrigerant states as function of time.
speed is held constant, but more cooling is required, the system has to generate higher cooling capacity in order to extract more heat through the evaporator, resulting in higher evaporation temperature and, hence, higher temperature of chilled water.
Fig. 3. Tch as function of motor speed and percent load.
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The open-loop simulation results presented above give the transient response characteristics of the system such as steady-state time. The developed model is useful in assessing the range of design parameters under various load conditions.
3. SUB-OPTIMAL ON±OFF CONTROL STRATEGIES
For the system dynamics given by Equations 1)±(20, an optimal control problem can be formulated and solved numerically to determine optimal control trajectories. Such methods are widely used (Braun et al., 1989). Diculties arise with regard to computational time, convergence and concerns about global optimality of the solution. These concerns and hardware costs limit their application on real systems. To this end, the objective of this study is to develop optimal or sub-optimal on±o control strategies. In order to develop sub-optimal on±o control strategies a two-tier scheme is proposed. First, a sub-optimal on±o control strategy based on a reduced-order CWC system is designed. The strategy so designed is then used to simulate the full order CWC system described by Equations 1)±(20 and the results are compared. In the following section, the reduced order model equations are developed, the optimal on±o control problem is formulated and a solution technique is described. 3.1. Construction of reduced-order model If Tch is the chilled water temperature and Cch is the storage capacity, the energy balance equation on the storage tank (Fig. 1) is written as Cch
dTch ÿU3 U3max COP Um Ummax Cwat T1 ÿ Tch dt
21
where the rate of energy stored in the storage tank is equal to the rate of energy extracted from the chilled water U3U3max COP and the heat gains from the return water UmUmmaxCwat(TAÿTch). Um in the equation is the normalized mass ¯ow rate of chilled water supplied to cooling coil, and U3 is the normalized input energy to the chiller. The coecient of performance (COP) of the chiller is modelled as (Jian, 1996). T1 ÿ Tch 22 COP COPmax ÿ 1 1 ÿ DTmax where TA is the sink temperature and D Tmax is the maximum temperature dierential the chiller is designed to work with. The cost function for minimizing the energy use is de®ned as Z tz a1 U3 dt a2 Tch ÿ Tchset 2 dt 23 J t0
The purpose of this function is in evaluating and minimizing the energy cost of operating the chiller (U3) and satisfying the setpoint temperature. The term a1U3 dt represents the amount of chiller energy input, a2(TchÿTchset)2 dt is used to minimize the temperature variations; a1 and a2 are weighting factors. Assuming the cooling load is known a priori, we propose a periodic solution for this problem to meet the following condition Tch tf ÿ Tchss 20:05 C
24
where Tchss is steady-state temperature. The constraint on the input capacity of the chiller U3 is taken to be 0 U3 U3max
25
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Fig. 4. Optimal strategy for load pro®le1. (a) Load pro®le1, (b) Tch response, (c) U3 pro®le, (d) cost function.
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3.2. Heuristic on±o control technique Consider a load pro®le as shown in Fig. 4. Since the load is time varying as shown in Fig. 4(a), we have to look for dynamic optimal solution for it. Based on heuristic ideas, we present a set of `optimal' switching modes consisting of fully on portion and unsaturated steadystate pro®les. The objective of the optimal switching strategy is that is should satisfy the boundary condition (24) and the minimum cost criterion. Figure 4(c) shows the optimal switching control pro®le for operating the chiller U3 (in solid line) corresponding to the load pro®le shown in Fig. 4(a). The method used to generate this control pro®le consists of the following two steps: (1) ®nding the steady-state optimal solution and (2) determining the lead time tld and on time ton of the system such that (a) the boundary condition is satis®ed and (b) the minimum cost criterion is met. 3.3. ON time search methodology A numerical search technique has to be used in order to determine the lead time and on time shown in Fig. 4(c). In the methodology used, the search problem is considered as a one dimensional search problem. In Fig. 4(a), since the load is increasing at 1.5 h, the chilled water storage has to be charged ahead of time so that water is available for supply at the set point temperature right at 1.5 h. With this assumption, U3 is turned on at time tld, and it is held at maximum value for period of `on' time ton, so that the chilled water temperature returns to its steady-state value at exactly 2.5 h. With tld and ton thus determined, the total cost was computed. 3.4. Test for optimality of control pro®les It is reasonable to ask whether the control pro®le shown in Fig. 4(c) (solid line) is optimal because there could be other solutions with dierent lead time tld. In order to answer this question, an extended search method was used. This time, we again use the numerical search method, but the problem is extended to a two variable search problem. The search for both t*ld and t*on was executed over a wide range, that is, from one hour before and half hour later from the point of application of load (1.5 h in Fig. 4a). For each time interval dt2, a search for the on time that minimized the cost was made. When starting the search procedure, an initial value for lead time was assumed and U3 is turned on, then for each small interval dt2, a search for ton was made such that the state variable returns to its steady-state value at 2.5 h, at this point U3 is set back to its steady-state value. If such ton is determined, one set of iteration has been completed for the lead time that was set at the beginning, and the corresponding cost value J is calculated for that iteration. The next round of searching will move ahead to the new lead time by adding a small amount dt1 to the initial value, and this process will continue until the lead time is equal to 2.0 h. While each step of searching is executed, the cost J was found. The result for optimality of the control method is shown in Fig. 4(d). In this ®gure, the cost J is plotted as a function of both lead time and on time. As it is shown in the ®gure, for each individual tld, there is a ton which causes Tch to go to steady-state value at 2.5 h. However, there is only one set of tld and ton that gives minimum cost Jmin (4.06). The switching control pro®le for U3 and the temperature Tch response for this optimal case are shown in Fig. 4(b, c) (solid line). 3.5. Further investigation of the on±o control pro®les In order to test the methodology of generating the heuristic on±o control pro®les for dierent load conditions, several runs were made by changing the load con®gurations. Since the ®rst load pro®le shown in Fig. 4(a) was referred to as case1, the new load is identi®ed here as case2 load pro®le as shown in Fig. 5(a). Although this load pro®le is somewhat dierent than the ®rst one, the same methodology for computing optimal on±o control solution will be used. For load pro®le shown in Fig. 5(a), again we expect to obtain the heuristic on±o control strategy for U3, which satis®es the boundary condition (24) and meets the minimum cost criterion expressed in (23). Figure 5(b) shows the cost J as function of tld and ton for case2. Results show that for lead time ranging between ÿ1.4 and 0.4 h, a convex function is obtained, and there is a point at
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Fig. 5. Chiller energy cost as function of lead time and on time for load pro®le2. (a) Load pro®le2, (b) cost function.
which the minimum cost Jmin accrues. The test results shown above demonstrate that the technique based on one-dimensional two variable search gave minimum cost under ¯uctuating load conditions. 3.6. Application of heuristic on±o control pro®les It is of interest to extend the heuristic on±o control technique described in the last section for a typical day operation. To this end, the ®rst step that is necessary is to determine the optimal switching solution over intervals of 1±2 h until a 24 h period is covered. 3.6.1. Multiple usage of heuristic on±o control technique for daily load. Consider a step change load pro®le as shown in Fig. 6(a). For the day with this load curve, known a priori, we wish to ®nd the heuristic switching control sequence for U3. In this case, the cost function for minimizing the 24 h pattern of energy use is de®ned as
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Fig. 6. Optimal unconstrained Tch and U3 responses. (a) Daily load case1 pro®le, (b) Tch response, (c) U3 response.
Z J
24 0
a1 U3 dt Tch ÿ Tchset 2 dt
26
The terms in Equation 26 are the same as those in Equation 23. We also expect the solution to this problem to satisfy the following boundary condition Tch 22 ÿ Tchss 20:05 C Moreover, the constraint on the chilled water temperature Tch is taken to be
27
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7:0 C Tch 9:0 C
381
28
For the daily load pro®le shown in Fig. 6(a), we propose an optimal unconstrained control solution by repeated application of the heuristic on±o control technique, more speci®cally, once every 2 h, between 10 and 22 h. In this way, the control solution for U3 which satis®es boundary condition (27) and meets the minimum cost criterion (26) was found. Meanwhile, the minimum cost Jmin for the day would be computed by integration of minimum cost values of each searched interval. Figure 6(b, c) presents the on±o control pro®le of U3 and resulting chilled water temperature Tch. As we see in the Fig. 6(b), the temperature Tch is beyond the constraint
Fig. 7. Open loop responses of Tch for three load cases. (a) Tch response at Um=0.3, (b) Tch response at Um=0.5, (c) Tch response at Um=0.8.
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W.-L. Jian and M. Zaheeruddin
limit (28). However, the magnitude of net overshoot Tos is found to be relatively small (0.35± 0.588C), and these overshoots occur between 12 and 18 h, which indicates that during the high load period, direct application of the control technique developed earlier could lead to some overshoot. It is possible to search for a new lead time to reduce the overshoots. It was found that one or two trials were enough to get a constraint solution with minimal or no overshoot in Tch. 3.7. Simulation of switching control strategies on full-order system In the above we have presented a methodology by which optimal on±o control strategies for a chilled water system can be computed. However, to illustrate the technique we chose a reduced-order model. It is of interest to see if the optimal on±o control strategy determined from the reduced-order model can be successfully simulated on the full-order model developed in Section 2. Before doing this it would be necessary to make sure that the reduced-order model behaves in a similar way to the full-order model. To do this, ®rst we need to make sure that the
Fig. 8. Comparison of Tch responses between two models for load pro®le2. (a) Load pro®le2, (b) Tch response.
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parameters in the reduced-order model are properly chosen so that the open loop responses are identical or close to that of the full-order model. The resulting open loop responses of chilled water temperature Tch for both models are shown in Fig. 7(a±c), for three dierent loads Um=0.3, Um=0.5 and Um=0.8. It is apparent that steady-state time is the same for both models. However, the trajectories of Tch are somewhat dierent, and for larger load Um=0.8, the dierence between the two Tch curves is more evident. This is because the full-order model is large and complex as such the system characteristics are eected by several parameters compared to the reduced-order model. Therefore, it can be considered that both models are close enough for the purpose of designing optimal on±o control using the reduced-order model.
Fig. 9. Comparison of Tch and energy input response of two models for daily load (case1). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
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3.7.1. Results. Simulation results are presented for load pro®le shown in Fig. 8(a). First the optimal on±o control strategy was found by using the reduced-order model. Then this optimal on±o control sequence was used to simulate the full-order system response. The resulting Tch pro®les were compared. We can see from the ®gure that the temperature Tch from full-order model returns to the steady-state value exactly at 2.5 h, thus satisfying the boundary condition (24). In addition, it is found that when load remains low, say during period of 0±1.5 h and 2.5± 3.5 h, the Tch response of the full-order model is quite close to that of reduced-order model; however, when load increases, the temperature Tch of the full-order model is obviously lower than that of the reduced-order model. This could be explained in that the full-order model was
Fig. 10. Comparison of Tch and energy input response of two models for daily load (case2). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
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Fig. 11. Comparison of Tch and energy input reponses of two models for daily load (case3). (a) Daily load case1 pro®le, (b) Tch responses, (c) control input responses.
developed more thoroughly and is considered to be a more accurate representation of system dynamics. The maximum temperature dierence in Tch between the two models is about 0.68C for load pro®le, shown in Fig. 8(a). The simulation results for the daily load are shown in Fig. 9Fig. 10Fig. 11(a±c)±11(a±c). It can be noted that by implementing the near-optimal on±o control pro®les on the full-order model, the response of Tch is basically kept close within the constraint. The comparison of temperature Tch pro®les between the two models is also given in the ®gures. They indicate that both Tch curves look very close and the largest dierence between them is 0.15±0.258C. These results show that the heuristic on±o control methodology developed in this study is simple and near optimal, and can be used for operation of chilled water cooling systems with storage.
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4. IMPLEMENTATION ISSUES
From the past history of load patterns and the anticipated load for the next day, the optimal lead time, switching times, and the resulting chilled water temperature pro®le can be computed ahead of time by using the methodology described in Section 3. In terms of implementation, it would be easier to track the chilled water temperature pro®le by using a two-position controller and a timer. The clock enables the controller to start the compressor at the computed lead time of the day. Thereafter, hourly optimal high and low limits of the chilled water temperature could be read from a reference table for eecting the on±o control sequence. Since the predicted load is likely to be dierent from the actual load, it may be necessary to ®ne tune the high and low limits somewhat, based on the actual load changes. 5. CONCLUSIONS
A dynamic model of a chilled water cooling system with storage useful for design and control analysis has been developed. The development of a sub-optimal on±o control strategy based on a reduced-order model and the simulation of on±o control strategies on the full-order chilled water cooling system have been studied. The results show that: (1) the sub-optimal on± o control strategy developed based on one dimensional search method has shown to be `optimal' when the chiller system is operating under time varying load, (2) the daily chilled water temperature response shows that the multiple application of the unconstrained optimal on±o control strategy with some ®ne tuning can give good control of chilled water temperature, (3) the results of simulation of the sub-optimal on±o control pro®le on the full order model show that, for the short period load con®gurations, the maximum dierence in the chilled water temperature between full and reduced order model are 0.15, 0.5, and 0.38C; for the daily load pro®les the dierences are 0.2, 0.18, and 0.18C. Being simple and sub-optimal, the on±o control technique can be successfully applied to control chilled water cooling systems. AcknowledgementsÐThis work was supported by funds (OGP0036380) from the Natural Sciences and Engineering Research Council of Canada.
REFERENCES 1. J. E. Braun, S. A. Klein, W. A. Beckman and J. W. Mitchell, Methodologies for optimal control of chilled water systems without storage. ASHRAE Transactions 95(1), 652±662 (1989). 2. J. Chi and D. Didion, A simulation model of the transient performance of heat pumps. International Journal of Refrigeration 5(3), pp. 176±184 (1982). 3. Dhar, M., Transient analysis of refrigeration system, Ph.D. Thesis. Purdue University (1978). 4. Gear, C. W., Numerical initial value problems in ordinary dierential equations. Prentice-Hall (1971). 5. Jian, W. L., Dynamic modelling and on±o switching control of a chilled water cooling system with storage, Master's thesis. Centre for Building Studies, Concordia University, Montreal, Canada (1996). 6. Yasuda, H. and Touber, S., Simulation model of a vapour compression refrigeration system. Laboratory for Refrigerating Engineering and Air Conditioning Rept. No. WTHD 133. Department of Mechanical Engineering, Delft University of Technology, The Netherlands (1982). 7. M. Zaheer-uddin and J. C. Y. Wang, Start-stop control strategies for heat recovery in multi-zone water-loop heat pump systems. Heat Recovery Systems & CHP 12(4), (1992). 8. M. Zaheer-uddin, R. E. Rink and V. G. Gourishankar, Heuristic control pro®les for integrated boilers. ASHRAE Transactions 96(2), 205±211 (1990).