1、1 I I n t e i- n a tio n a 1 J o u i. n al of H e at in g,Veii t i la t in g , Air-conditioning and Refrigerating Research HVAC 6r, . The average cost for the service cycle can then be expressed as a The analysis can be further simplified by assuming that the longest period for the natural driving c
2、onditions %t) is one year. In this study, typical meteorological year (TMY) Hall et al. 1978) weather data were used in the simulations and the typical years weather pattern reoccurred each year. With this simplification, the table only requires n,-n, elements (K31 15 n, 1 I k I n,) and the duration
3、 of the service cycle Tc is an integer multiple of one year. ASHRAE TITLE*IJHVAC 2-1 9b 0759b50 05LBB37 234 = VOLUME 2. NUMBER 1. JANIJARY 1996 7 Given a model for determining Px( t)J( t) needed to calculate Kfi 1 2 1 5 ns, 1 5 k I %), the cost ratio CJC, and the minimum service interval At, the opt
4、imization problem can be defined as minimizing JI with respect to (Te, Ne, tll and the number of time stages between service tasks -8. This is an N, + 2 dimensional optimization problem. The minimum service internal At can be the minimum reaction time of the service orga- nization or a sufficiently
5、small quantity such that the minimum cost no longer depends on its value. In the latter case, At would be decreased until the minimum cost became insensitive to At. For the cleaning of heat exchangers, a minimum service interval of 1 month is a realistic and sufficiently accurate interval for schedu
6、ling service that was used in this study. The optimal maintenance scheduler minimizes the lifetime costs of energy and ser- vice while maintaining comfort, safety, and environmental protection as constraints. These constraints are maintained by adding artificially high costs to the tabulated energy
7、usage IC: for operating conditions that would result in constraint violations. The numerical solution to the optimization problem is accomplished using a combina- tion of two numerical techniq%e:. Given values of T, N, t,. the optimal set of time stages between service tasks ST = S.t;,St;, ., ST_, i
8、s determined using dynamic programming (Bellman 1957). An outer loop, containing the dynamic programming solu- tion. is then used to find the optimal values of the other three quantities: q, NE, T; . Dynamic Programming The use of dynamic programming for solving an optimal equipment replacement prob
9、lem is discussed in Jardine (1973. section 4.41. Dynamic programming determines a global minimum in an efficient manner for this type of problem by taking advantage of the fact that the number of possible solutions is restricted by the sequential nature of the decision process. Ra0 1984) offers a go
10、od description of dynamic programming. Figure 2 is a black box diagram illustrating the sequence of decisions required to solve for the optimum placement of N, service tasks among n, time stages, starting at time stage T. There are Ne - 1 decisions that have to be made and each one is referred to as
11、 a decision stage. The state information passed between decision stages is the time stage that service was last performed ti. for the ith stage). The decision variable is the number of time stages to wait until the next service cazi, for the ith stage). The cost of each decision stage (ci), except t
12、he last, is the energy used while running the unit forari time stages starting at q plus the service cost divided by the energy cost. The cost of the last decision stage ( CNc - 1 ) also includes the energy consumed while running the unit back to the first decision stage plus the cost of the first s
13、enrice task normalized by the energy cost. Finally, the output of each stage is the time of the last service plus the run time for that decision stage = T +a ti). The sum of the costs of the decision stages is given by Dynamic programming is appropriate for this optimization problem because the per-
14、 formance of the vapor compression equipment does not depend on the chain of events that led to the most recent servicing. Figure 2 illustrates this point in that the decision to do service only requires knowledge of the time of the previous service task repre- sented by the state variable passed be
15、tween decision stages). This property of the opti- mal servicing problem guarantees an optimal solution without the need for testing all possible service schedules. Figure 3 illustrates an example dynamic programming problem for optimal mainte- nance scheduling. The abscissa contains the decisions s
16、tages and the ordinate contains ASHRAE TITLEWIJHVAC 2-1 96 0759650 0538838 170 H WAC- RESEARCH 8 4 Figure 2. Dynamic programming places iv, - 1 service tasks to minimize Ji the time stages when service can be done. In this example, there are n, = 12 time stages in a T, = 1 year service cycle (i.e se
17、rvice opportunities are at monthly intervals and the cycle repeats each year). There are N, = 4 service tasks to place and the time stage of the first task is specified as il = 3. The open circles indicate the available opportunities for doing service at each decision stage. Not all values are allow
18、ed for each stage because ii+l is restricted by definition to follow ii. This reduces the number of possible decisions to test, thereby reducing computation time. In general, the number of avail- able time stages at each decision stage is nc - N, + 1. W u U 2 3 4 1 2 Decision Stage Figure 3. Optimal
19、 trajectory through domain for an example problem The example problem of Figure 3 is solved by starting at decision stage 1 and pro- gressing backwards one decision at a time until decision stage 1 is reached again and the cycle repeats itself. Beginning at decision stage 1 where service is performe
20、d at time stage 3). the costs of getting there from all nine allowed time stages in decision stage 4 are computed and stored. Next, all possible time stages in decision stage 3 are considered. For each of these nine time stages, the costs of moving to each possible time stage in decision stage 4 are
21、 computed. Then. the minimum costs-to-go from each of the nine time stages at decision stage 3 to decision stage 1 are computed and stored with the associated path. This process is repeated until decision stage 1 is reached. ASHRAE TITLEMIJHVAC 2-1 96 0759650 0518839 007 where g I 4.2 4.4 4.6 4.8 5
22、tf (years) 1.5 1 .o 0.8 - ;ii . % 0 ._ 0.6 e P 2 6 v al I 0.4 .8 07 0.2 20.0 (b) PLF, = 0.80 Figure 7. Portion of constrained only solution plotted in Figure 6 included and are referenced to the right axes. The service rate is the average number of services per year computed over one service cycle.
23、Dotted vertical lines indicate when the service rate changes. The primary causes of the irregular nature of the cost curves are discontinuities due to sudden changes in service costs when the service rate changes followed by changing energy costs as the fouling time changes for a constant service ra
24、te. Secondary causes of irregular behavior in the cost curves are discrete changes in the month service is performed with the same service rate) and changes in the constraint violation that triggers the service. Both of these secondary features are included in these figures. For example, when PLF, =
25、 0.80 and the service rate equals one service every three years, the total costs change when the service time changes from June month 6) to September month 9). For the case of PLF, = 0.55 and the ser- vice rates equal to one service every 1. 2, and 3 years, the operating costs change rap- ASHRAE TIT
26、LE*IJHVAC 2-1 96 m 0759650 0518849 TSb m VOLUME 2. NUMBER 1. JANUARY 1996 19 idly when the service time changes from June to April because the constraint violation switches from comfort to low suction pressure. Cost Versus t * for Condenser Fouling f Figure 8 compares the combined costs of energy an
27、d service for different service schedules versus the fouling rate of the condenser. The trends and savings associated with optimal maintenance scheduling are similar to those reported for evaporator foul- ing. However, the scale of the vertical axis (extra cost) is larger due to the higher cost for
28、condenser cleaning ($100) as compared to evaporator fouling ($60) resulting in larger absolute savings for optimal maintenance scheduling. In contrast to the results of Fig- 1 I I 1 2 3 4 5 6 ti (years) (a) PLF, = 0.55 U O 0.3 - - Once/Season + Twice/Season o Optimal - - Simplified 0.451 i I 0.051 1
29、 I 1 2 3 4 5 6 O O t (years) (b) PLF, = 0.80 Figure 8. Additional annual operating costs versus for condenser fouling ASHRAE TITLE*IJHVAC 2-1 96 = 0759b50 0538850 778 HVAC = 2.75 years for PLF, = 0.55 and = 1.95 years for PLF, = 0.80. When C, was varied, PLF, was fixed at 0.55 and when PLF, was vari
30、ed, C, was fixed at $100. The results of Figure lO(a) are similar to those of Figure 9(a). For low service costs, the optimal and regular schedules give similar costs since they all provide enough service and there is a small penalty for over servicing. As service costs rise. the regular schedules c
31、ost more because they are providing excessive ser- vice. In contrast, the constrained schedule under services, thereby resulting in larger cost penalties at low service costs. Unlike evaporator fouling, the costs of regular and optimal service schedules are not independent of PLF, for a fKed fouling
32、 rate. When the equipment size becomes larger (smaller PLF,), the condenser fans run less and fouling occurs more slowly (clock time), thereby decreasing the costs for all the schedules. The optimal schedule has the ability to reduce the service rate to less than once per year as the unit size incre
33、ases, causing the costs of the once/year and optimal schedules to separate for low PLFm. Effect of the Fouling Model For any point in time, an estimate of the optimal service time from the simplified ser- vice scheduler is based on the assumption that the current dependence of fouling on runtime wil
34、l not change in the future. Under this assumption, the performance of this scheduler is extremely good, as demonstrated in Figures 6 through 10. This section investigates the effect of using a fouling model that changes with time. The alternative fouling model consists of replacing portions of the c
35、onstant fouling rate with randomly placed impulses while maintaining tf as the characteristic fouling time. The impulses model sudden unanticipated accumulations of dirt on the heat exchangers. A mathematical description of the fouling model is: ASHRAE TITLE*IJHVAC 2-1 96 m 0759b50 0518852 5VO m 22
36、HVACi r- - aipha=l.O - aipha=0.7 - alpha=0.4 - aipha=0.0 6000 7000 23 (b) n = 3 Figure 11. Fouling state versus runtime hours pletely comprised of a series of impulses, and when a = O, the model is completely lin- ear. The parameter n is the number of impulses per calendar year. The impulses are ran
37、domly placed throughout the year, but reoccur at the same time each year. Figure 11 shows the fouling state f as a function of runtime for a fouling time of tf = 7830 hours (number of evaporator runtime hours in three cooling seasons). The vertical lines separate the cooling seasons. Figure 1 1 (a)
38、illustrates the dependence on n for a = 0.8 and Figure 1 l(b) illustrates the dependence on ci for n = 3. Ail fouling models attain complete fouling in the characteristic time tf The more nonlinear models are characterized by a near 1 and small n (e.g., n = 1). In this case, fouling occurs as a 24 H
39、VAC&R RECEARCH series of large step changes. As n grows larger, the evolution of fouling approaches a lin- ear model. regardless of a. Table 1 summarizes the costs for different fan control strategies and building capac- ity-to-load ratios for both evaporator and condenser fouling. The costs are the
40、 percent additional costs relative to optimal maintenance scheduling (actual minus optimal cost over the optimal cost times 100) averaged over 11 values of cx between O and 1 and 5 values of n between 1 and 5. The fan control strategies are call for cooling (CFC), mean- ing that the fan operates onl
41、y when the compressor is operating, and occupancy sched- ule (OCC), meaning that the fan runs during occupied periods for ventilation. For the cases considered. the simplified scheduler provided near-optimal operating costs. For evaporator fouling with PLF, = 0.80 (little extra capacity), the constr
42、ained solution also provided performance comparable to the optimal schedule. Table 1. Performance Comparisons Between Different Service Schedules YO Additional Cost Relative to Optimal Fouling Type Control PLF, Simplified Once/Season Constrained Twice/Season Condenser CFC 0.55 0.56 2.68 5.60 11.50 C
43、ondenser CFC 0.80 0.53 1.85 4.42 9.93 Evaporator OCC 0.55 0.46 2.11 3.74 7.88 Evaporator OCC 0.80 0.46 2.1 1 0.65 7.88 Evaporator CFC 0.55 0.36 1.81 2.61 7.58 Evaporator CFC 0.80 0.80 1.21 0.21 6.63 CONCLUSION This paper demonstrated that there is a significant opportunity for cost savings asso- cia
44、ted with optimal scheduling of condenser and evaporator maintenance. It was found that optimal service scheduling reduced lifetime operating costs for a rooftop air rondi- tioner by as much as a factor of two over regular service intervals. and 50% when com- pared to constrained only service. For pr
45、actical implementation, a simple near-optimal algorithm was developed for estimating optimal service times. In contrast to the optimal solution, this approach does not require on-line forecasting or numerical optimization and is easily implemented with a micro-controller. Over a wide range of cases
46、tested, the near-optimal algorithm gave operating costs that were within 1% of the optimal results. This study was performed for a small rooftop air conditioner. Greater opportunities for cost savings may be possible for larger equipment such as water chillers or electric power generating plants. Th
47、ese systems also experience performance degradations, but have significantly higher service and energy costs. Future work should extend the approaches described in this paper for determining optimal and near-optimal service times to these other systems. Implementation of the near-optimal maintenance
48、 sched- uler should also be tested in the laboratory under controlled conditions. NOMENCLATURE cs service cost $) HU,) h( t,) dt, (kWh1 JO J1 = JO/& simplified cost function (kWh/year) ce energy cost ($/kWhl J2 =Ji% C mass flux (kg/(s.m2) 53 cost function with fewest assump- tions ($/year) dynamic p
49、rogramming cost function (kWhl simplified cost function assuming the time between service tasks is fixed (kWl simplified cost function subtracting out the expected power consumption with no fouling kW) ASHRAE TITLErIJHVAC 2-1 9b 0759b50 0518855 25T VOLUME 2, NUMBER 1. JANUARY 1996 energy consumed starting at time stage I after servicing and operating for k time stages (kwh) number of service tasks during equipment lifetime (dimensionless) number of service tasks during ser- vice cycle (dimensionless) hours of fan operatio
