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A General control algorithm for cooling towers in cooling plants with electric and/or gas-driven chillers.

Received September 13,2006; accepted February 23, 2007

INTRODUCTION

A chiller plant often consists of multiple chillers, multiple condenser water pumps, and multiple cooling towers as depicted in Figure 1. In some situations, it is economical to employ a mix of chillers that are "powered" by electricity and natural gas (e.g., absorption or engine-driven). This is termed a hybrid chiller plant. The major advantage of using natural gas chillers in hybrid central chiller plants is a reduction in peak electrical demand and on-peak energy usage, which can reduce overall operating costs. The electric demand cost can often account for about half of the total air-conditioning bill.

[FIGURE 1 OMITTED]

Proper control of cooling tower fans in cooling plants can have a significant impact on operating costs. For all electric plants, optimal control of cooling fans has been shown to result in significant (e.g, 5%--15%) savings in plant energy costs as compared with typical strategies that are employed (see Sud [1984], Lau et al. [1985], Hackner et al. [1985], Klein et al. [1988], Braun [1988], Braun et al. [1989a, 1989b], ASHRAE [2003]). The savings for plants employing absorption chillers can be even larger because of higher heat rejection requirements. For instance, Koeppel et al. (1995) simulated optimal control of tower fans and condenser pumps for a cooling plant having a double-effect absorption chiller and determined a 20% reduction in costs compared to using fixed speeds with a tower bypass control to maintain a constant cooling tower water supply.

Although there is a large body of literature related to supervisory control for all-electric plants, there is very little literature on supervisory control of absorption, engine-driven, and hybrid chiller plants. Braun and Diderrich (1990) developed an algorithm for cooling tower fan control for all-electric plants that is included in the 2003 ASHRAE Handbook--HVAC Applications (ASHEAE 2003). However, this strategy is not appropriate fro hybrid plants because it is based on minimizing input energy usage rather than cost.

Koeppel et al. (1995) developed a simplified strategy for cooling tower fan control for absorption cooling plants that involves the determination of a linear relationship between a setpoint for cooling tower supply water temperature and ambient wet-bulb temperature. Optimization results were used to determine a simple linear model for a case study involving a single double-effect absorption chiller. The application of the simple strategy resulted in savings that were nearly identical to the optimization results. However, it's not obvious how linear control relationships would be determined in practice and how to apply this method to hybrid plants.

This paper develops a general algorithm for control of cooling tower fans for cooling plants that have any combination of electric and natural-gas chillers. The development follows an approach that is similar to the development of Braun and Diderrich (1990). The control method is evaluated through comparisons with optimal control for a range of different cooling plants and operating conditions using a simulation tool. Optimal control means that the control is based on minimization of an operating cost function that incorporates perfect information about the plant. A nonlinear optimization was performed to determine the performance for the benchmark optimal control using the simulation tool. The development of the near-optimal control algorithm used several simplifying assumptions and heuristics in order to determine analytical expressions for cooling tower control. In this context, near-optimal control implies that the strategy has performance that is close to that associated with optimal control.

DEVELOPMENT OF A NEAR-OPTIMAL COOLING TOWER CONTROL ALGORITHM

Chiller plants, such as that depicted in Figure 1, typically have multiple cooling towers with fans that have multiple speeds of operation. In general, optimal control of cooling tower fansresults from a trade-off in the cost of operating the chillers and cooling tower fans. The energy consumption of a chiller is sensitive to the condenser water temperature, which is affected by the cooling tower control. Increasing the tower airflow reduces the chiller energy requirement but at the expense of an increase in fan power consumption. For a given set of conditions, an optimal tower control exists that minimizes the sum of the chiller and cooling tower fan power.

Braun and Diderrich (1990) described how the determination of optimal tower fan control can be separated into two parts: tower sequencing and optimal airflow. For a given total tower airflow, optimal tower sequencing specifies the number of operating cells and the fan speeds that give the minimum fan power consumption. Once the tower sequencing is specified, then the optimal airflow can be determined by analyzing the trade-offs between the costs of operating the chiller and the fan.

This section presents the development of an algorithm for near-optimal control of cooling towers that is based upon a combination of heuristic rules for tower sequencing and an open-loop control equation derived from a detailed analysis.

Optimal Fan Sequencing

Simple relationships exist for the best sequencing of cooling tower fans for towers having multiple cells as capacity is added or removed. When additional tower capacity is required, Braun et al. (1989a, 1989b) have shown that in almost all practical cases, the speed of the tower fan operating at the lowest speed (including fans that are off) should be increased first. Similarly, for removing tower capacity, the highest fan speeds are the first to be reduced. This leads to the following general rules for sequencing of tower fans:

1. All Variable-Speed Fans: Operate all cells with fans at equal speeds.

2. Multi-Speed Fans: Increment lowest-speed fans first when adding tower capacity. Reverse for removing capacity.

3. Variable/Multi-Speed Fans: Operate all cells with variable-speed fans at equal speeds. Increment lowest-speed fans first when adding tower capacity with multi-speed fans. Add multi-speed fan capacity when variable-speed fan speeds match the fan speed associated with the next multi-speed fan increment to be added.

Criteria for Optimal Tower Airflow

Most cooling towers utilize single-or two-speed fans, such that the optimization problem is discrete rather than continuous. However, for the purpose of estimating the control parameters, it is sufficient to consider the flow as being continuously adjustable. Consider the problem of determining the optimal tower control for continuously adjustable tower airflow. The minimum combined chiller and tower fan cost occurs at a point where the rate of change of cost with respect to changes in tower airflow is equal to zero, or

[dC.sub.twr]/[d[gamma].sub.twr] = [dC.sub.ch]/[d[gamma].sub.twr], (1)

where [C.sub.twr]and [C.sub.ch] instantaneous energy costs for the cooling tower fans and chillers and defined as ratio of the airflow to the maximum possible airflow with all cells operating at maximum speed. In order to solve for the optimal tower control, it is necessary to develop a functional relationship for the sensitivities of chiller and cooling tower costs to tower airflow.

Chiller Cost Sensitivity to Tower Airflow

The rate of change of chiller cost with respect to tower airflow may be expressed as

[[dC.sub.ch]/[d[gamma].sub.twr]] = [[dC.sub.ch]/[dT.sub.cwr]].[[dT.sub.cwr]/[d[gamma].sub.twr]], (2)

where [T sub.cwr] is the condenser water return temperature. The rate of change of total chiller cost with respect to condenser water return temperature is

[[dC.sub.ch]/[dT.sub.cwr]] = [[N.sub.ch].[summation over (i-1)] [ER.sub.i]].[[dE.sub.[ch,i]]/[dT.sub.cwr]], (3)

where [E sub [ch,i]] is the rate of energy input and [ER.sub i] is cost per unit energy input (heat or electricity) for the ith chiller. The rate of change of energy input with respect to condenser water temperature written as

[[dE.sub.[ch,i]]/[dT.sub.cwr]] = [[S.sub.[c,h,i]].[COP.sub.[rated,i]]/[COP.sub.i].[E.sub.[ch,rated,i]]], (4)

where COP is the chiller coefficient of performance (ratio of cooling to energy input), [COP.sub.rated] is the chiller COP at the chiller design conditions, [PLR.sub.ch] is the chiller part-load ratio (load relative to rated capacity), and [S.sub.ch] is the sensitivity of the chiller input energy to changes in condenser water temperature given as

[S.sub.[ch,i]] = 1/[[E.sub.[ch,i]].[dE.sub.[ch,i]]]/[dT.sub.cwr]., (5)

For the purpose of determining tower airflow, it is reasonable to assume that the factor [S.sub.ch]([COP.sub.rated] /COP) is constant for a given chiller over different operating conditions. With this assumption, Equation 4 can be simplified to

[[dE.sub.[ch,i]]/[dT.sub.cwr]] = [[S.sub.[ch,rated,i]].[PLR.sub.[ch,rated,i]].[E.sub.[ch,rated,i]], (6)

Where [S.sub.[ch,rated,i]] is evaluated at the chiller design conditions.

The next step is to develop a relationship for the effect of the tower control on the condenser water return temperature. From an effectiveness model for the thermal performance of a cooling tower (Braun et al. 1989c), the condenser water return temperature may be expressed as

[T.sub.cwr][equivalent][T.sub.wd]+[[Q.sub.twr]/[[epsilon].sub.[a,twr]][m.sub.[a,twr]][c.sub.s]], (7)

where [T.sub.wb] is ambient wet-bulb temperature, [Q.sub.twr]. is the cooling tower heat rejection rate, [m.sub.[a,twr]] is the tower air mass flow rate, [epsilon.sub.[a,twr]].is the an air-side effectiveness for heat and mass transfer within the tower, and [C.sub.s] is a property of the air-water vapor mixture termed the saturation specific heat.

Equation 7 is simplified by relating the performance to design conditions, assuming that the tower effectiveness and saturation specific heat

Equation 7 is are constants and utilizing the definitions for relative tower airflow, and tower approach and range.

[T.sub.cwr] = [T.sub.wb] + [Q.sub.twr]/[[epsilon].sub.[a,twr]][m.sub.[a,twr]][c.sub.s] [[epsilon].sub.[a,twr,rated]][m.sub.[a,twr,rated]][c.sub.[s,rated]](T.sub.[cwr,rated]-[T.sub.[wb,rated]])/[Q.sub.[twr,rated]] = [T.sub.wb] + [Q.sub.twr]/[m.sub.[a,twr]]/[m.sub.[a,twr,rated]][Q.sub.[twr,rated]] [[epsilon].sub.[a,twr,rated]][c.sub.[s,rated]]/[[epsilon].sub.[a,twr]][c.sub.s]([a.sub.[twr,rated]] + [r.sub.[twr,rated]] ~[T.sub.wd] + 1/[[gamma].sub.twr] [Q.sub.twr]/[Q.sub.[twr,rated]](a.sub.[twr,rated] + r.sub.[twr,rated]), (8)

where [a.sub.[twr,rated]] and [r.sub.[twr,rated]]. are the tower approach and range evaluated at the rating condition with the tower operating with maximum tower airflow at the chiller design load. The approach is the difference between the condenser water supply and ambient wet-bulb temperatures ([T.sub.cws]-[T.sub.wb]), whereas the range is the difference between the condenser water return and supply ([T.sub.cwr] - [T.sub.cws]).

The ratio of the tower heat rejection to the design heat rejection can be approximated as

[Q.sub.twr]/[Q.sub.[twr,rated]] = [N.sub.ch].[summation over (i=1)][PLR.sub.[ch,i]].[f.sub.[twr,rated,i]], (9)

Where

[f.sub.[twr,rated,i]] = [Q.sub.[cw,rated,I]]/[Q.sub.[twr,rated]], (10)

and where [Q.sub.[cw,rated,i]] is the heat rejection from ith chiller to the condenser water at the rating condition.

Equations 8 and 9 lead to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Combining Equations 2, 3, 6, and 11 results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Cooling Tower Fan Cost Sensitivity to Tower Airflow

The sensitivity of the tower fan costs to changes in control depends upon the type of fans and motors employed. For variable-speed drives and with each tower cell operating at equal flows, the fan power varies approximately with the cube of the airflow. For multi-speed fans and with an optimal sequencing strategy, the fan power varies as a piecewise linear function that approaches the variable-speed relationship as the number of speed settings increases. For single-speed fans, the fan power increases as a single linear function of the total tower airflow.

For a system with discrete tower fan settings, it is adequate to assume a continuous variation in order to perform the optimization and determine the discrete control that is closest to the estimated control. For single-speed fans, the power is assumed to be a continuous linear function. For two-speed fans, performance data suggest that a squared relationship for power consumption as a function of airflow is adequate. For three-speed (or more) fans, a cubic variation in power consumption with airflow is sufficient.

Thus,for varible-speed or three-speed fans, the sensitivity of the fan power consumption to control is assumed to be

[dC.sub.twr]/[d[gamma].sub.twr] = 3.[[gamma].sub.twr.sup.2].[ER.sub.e].[P.sub.[twr,rated]], (13)

While for two-speed fans

[dC.sub.twr]/[d[gamma].sub.twr] = 2.[[gamma].sub.twr].[ER.sub.e].[P.sub.[twr,rated]], (14)

and for single-speed

[dC.sub.twr]/[d[gamma].sub.twr] = [ER.sub.e].[P.sub.[twr,rated]], (15)

where E[R.sub.e]. is cost per unit electricity input and [P.sub.[twr,rated]] is the rated fan power with the fans running aT capacity.

Optimal Tower Airflow

Substituting Equations 12 and 13 into Equation 1 and solving for [[gamma].sub.twr] gives the following relationship for near-optimal control of cooling towers with three-speed or variable-speed fans.

[[gamma].sub.twr] = [{1/3([a.sub.[twr,rated]] + [r.sub.twr,rated]) [[N.sub.ch].summation over (i = 1)] ([PLR.sub.ch,i][f.sub.[twr,rated,I]])[[N.sub.ch].summation over (i=1)]([PLR.sub.[ch,I]][S.sub.[ch,rated,i]) [[ER.sub.i][E.sub.[ch,rated,i]]]/[[ER.sub.e][P.sub.[twr,rated]]])}.sup.[1/4]], (16)

For two-speed fans, Equation 14 is used instead of Equation 13 and the near-optimal control

[[gamma]sub.twr] = [{1/2([a.sub.[twr,rated]] + [r.sub.[twr,rated]]) [[N.sub.ch].summation over (i=1)] ([PLR.sub.[ch,i]][f.sub.[twr,rated,i]])[[N.sub.ch].summation over (i=1)] ([PLR.sub.[ch,i]][S.sub.[ch,rated,i]] [[ER.sub.i][E.sub.[ch,rated,i]]]/[[ER.sub.e][P.sub.[twr,rated]]])}.sup.[1/3]], (17)

while for single-speed fans,

[[gamma].sub.twr] = [{([a.sub.[twr,rated]] + [r.sub.[twr,rated]]) [[N.sub.ch].summation over (i=1)] ([PLR.sub.[ch,i]][f.sub.[twr,rated,i]])([PLR.sub.[ch,i]][S.sub.[ch,rated,i]] [[ER.sub.i][ER.sub.[ch,rated,i]]]/[[ER.sub.e][P.sub.[twr,rated]]])}.sup.[1/2]], (18)

The relative tower airflow is the ratio of airflow to the rated airflow with all fans operating at maximum speed. The factors that affect the optimal relative tower airflow are (1) the part-load ratio for each chiller, PL[R.sub.[ch,i]]; (2) the ratio of the operating costs per unit time for each chiller at its rating conditions to the rated cooling tower operating costs per unit time, (E[R.sub.i][E.sub.[ch,rated,i]])/(E[R.sub.e][P.sub.[twr,rated]]); (3) the sensitivity of individual chiller input energy requirement to changes in condenser water temperature at the chiller rating conditions, Sch,rated,i; (4) the ratio of the ith chiller heat rejection rate at its rating condition to the rated cooling tower heat rejection rate, [f.sub.[twr,rated,i]]; and (5) the sum of the tower approach and range at the tower rating condition ([a.sub.[twr,rated]] +[r.sub.[twr,rated]]). All of the factors in these expressions, except chiller part-load ratio and utility rates, are based on design information and do not require measurements.

The design approach to wet-bulb, [a.sub.[twr,rated]] is the temperature difference between the tower sump water and the ambient wet-bulb for the tower operating at its air and water flow capacity at the tower rating conditions. The rated range, [r.sub.[twr,rated]] is the water temperature difference across the tower at these same conditions. The sum of [a.sub.[twr,rated]] and [r.sub.[twr,rated]] is the temperature difference between the tower inlet and the wet-bulb and represents a measure of the tower's capability to reject heat to ambient relative to the system requirements. A small temperature difference results from a high tower heat transfer effectiveness or high water flow rate and yields lower condenser water temperatures with lower chiller energy consumption, resulting in a lower optimal tower airflow. Typical values for the design approach and range are 7[degrees]F and 10[degrees]F.

Chiller part-load ratio measures the load for each chiller and influences the optimal tower airflow in two ways. First, chiller loading influences the total heat rejection requirements of the cooling tower, which affects the optimal airflow. The optimal tower airflow increases with heat rejection requirement. The first summation on the right-hand side of Equations 16--18 characterizes this effect. The factor [f.sub.[twr,rated,I]] weights the individual chiller loadings according to their effect on the total heat rejection and is larger for chillers having greater design cooling capacity or lower design COP (e.g., absorption).

The chiller loading also influences the chiller to cooling tower cost ratio along with the cost ratio at the rating conditions. As the ratio of chiller to tower cost increases, it becomes more beneficial to operate the tower at higher airflows. If the tower airflow were free, then the best strategy would be to operate the towers at full capacity independent of the load. The cost ratio is typically higher for hybrid cooling plants than for all-electric plants because of lower chiller COPs and higher heat rejection requirements for gas-driven chillers.

The chiller sensitivity factor, Sch, is the incremental increase in chiller operating cost for each degree increase in condenser water temperature as a fraction of the power, or

(change in chiller input energy rate)

[S.sub.ch] = [(change in chiller input energy rate)/[(change in condenser water temperature) . (chiller input energy rate)] , (19)

If the chiller input energy rate increases by 2% for a 1 degree increase in condenser water temperature, then S is equal to 0.02. A large sensitivity factor means that the chiller input energy then S is equal to 0.02. A large sensitivity factor means that the chiller input energy rate is very sensitive to the cooling tower control, favoring operation at higher airflows. The sensitivity factor should be evaluated at design conditions using chiller performance data. Typically, the sensitivity factor is between 0.01 and 0.03[degrees][F.sub.-1]

PERFORMANCE EVALUATION

The performance of the near-optimal control algorithm was evaluated through comparisons with optimization results for typical hybrid cooling plants that were simulated using a tool described by Braun (2006). A brief review of the modeling approach and plant characteristics is provided in this section along with the performance evaluation.

Chiller Plant Modeling

The cooling plant simulation tool neglects energy storage effects and determines hourly costs cooling for a specified combination of chillers, condenser water pumps, and cooling any given hour, the cost of energy associated with operating the chiller plant is

[C.sub.pl] = ([P.sub.twr] + [P.sub.cwp])[ER.sub.e] + [[N.sub.ch].summation over (i = 1)] [E.sub.[ch,i]][ER.sub.i], (20)

where [P.sub.twr] is cooling tower fan power, [P.sub.cwp] is condenser water pump power, [E.sub.[ch,i]] is the rate of energy usage for the ith chiller (gas or electric), [N.sub.ch] is the total number of chillers (gas and electric), [ER.sub.e] is the cost per unit of electrical energy, and [ER.sub.i] is the cost per unit energy input (heat or electricity) for the ith chiller (gas or electric). The rate of energy consumption for the ith chiller is determined as

consumption for the ith chiller is determined as

[E.sub.[ch,i]] = [[gamma].sub.[ch,i]].[PLF.sub.i].[Q.sub.[ch,rated,i]]/[COP.sub.[rated,i]], (21)

where [[gamma].sub.[ch,i]] is a control variable (0 or 1) that indicates whether the chiller is operating or not, [Q.sub.[ch,rated,i]] is the rated chiller capacity, [COP.sub.[rated,i]] is the rated chiller COP, and [PLF.sub.i] is the part-load factor, defined as the ratio of energy usage to the value at the rating condition.

The chilled-water supply temperature and condenser water flow rate are assumed to be constant, and the chiller part-load factor is correlated in terms of the chiller load and entering condenser water supply temperature. Figures 2--5 show part-load factors (PLF) as a function of chiller part-load ratio (PLR) and entering condenser water temperature ([T.sub.cws]) for the four different chillers considered within the simulation tool: (1) electric centrifugal chiller with variable-speed motor, (2) electric centrifugal chiller with fixed-speed motor and inlet guide vane capacity control, (3) single-effect absorption chiller, and (4) engine-driven centrifugal chiller. These figures were generated by using correlations to manufacturers data. The PLR is the ratio of chiller load to a rated capacity. Not surprisingly, the variable-speed electric chiller has the best part-load performance, whereas the fixed-speed electric has the worst.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In order to simulate a cooling plant, rated cooling capacities ([Q.sub.[ch,rated]]) and COPs ([COP.sub.rated]) are specified for each chiller. The model assumes that all of the energy input to the chillers is rejected to the condenser water loop. The temperature of water leaving the condenser is determined from an energy balance on the condenser.

Individual condenser pumps are assumed to be dedicated to individual chillers and to provide rated flow rates. The total condenser flow rate and pump power are simply the sums of the rated values for the chillers that are operating. Water flows and power are scaled with chiller cooling capacity according to user-specified rating values for volumetric flow rate per unit rated cooling capacity (gpm/ton or L/s?kW) and pump power per unit volumetric flow rate (W/gpm or W?s/L).

An effectiveness model presented by Braun et al. (1989c) is used to represent the performance of cooling tower cells. The model considers the impact of air and water flow rates on heat and mass transfer rates using characteristics that are representative of commercial cooling towers. For simplicity in developing and evaluating control strategy heuristics, the cooling tower is modeled as a single cell having continuously variable airflow rate. The performance of a cooling tower that has multiple cells that are operating identically (e.g., same flows) is equivalent to the performance of a single larger cooling tower having the same total water and airflow rates.

There are two limiting cases that are considered for estimating the cooling tower fan power: (1) variable-speed fans and (2) single-speed staged fans. These two cases represent the upper and lower bounds for performance associated with a particular tower design. For variable-speed fans, the fan laws are employed and the power varies with the cube of the airflow. This characterizes the behavior of either a single large tower with a variable-speed fan or multiple smaller tower cells with variable-speed fans all operating at the same speeds. For staged fans, it is assumed that the fan power varies linearly with airflow. This characterizes the behavior of multiple tower cells having single-speed fans that are staged on and off.

The cooling tower size and design fan airflow rates and power consumption are scaled according to the plant heat rejection requirements. Rated airflows and power are specified in terms of volumetric airflow rate per unit condenser heat rejection rate at design (cfm/ton or L/s kW) and fan power per unit volumetric flow rate (W/cfm or W s/L).

For given cooling load, ambient air conditions, and plant control variables, the cooling plant model iteratively determines the condenser entering and leaving water temperatures that balance condenser and tower heat rejection. The primary ambient condition that influences plant cooling performance is the air wet-bulb temperature, whereas the dry-bulb temperature has a minor effect. The control variables are the sequencing and relative loadings for individual chillers and the relative tower fan speed or airflow.

Overall Plant Characteristics

Table 1 gives parameters used in system simulations to evaluate the performance of the near-optimal tower control algorithm. The design ambient conditions, performance ratings, and embedded component performance characteristics were used along with a specification of the number, type, and cooling capacities of chillers to determine the cooling tower design airflow and fan power requirements. For all cases considered in this study, two 500-ton chillers were employed. All six possible combinations of two different chillers were chosen from the four available types: (1) fixed-and variable-speed electric, (2) fixed-speed electric and absorption, (3) fixed-speed electric and engine, (4) variable-speed electric and absorption, (5) variable-speed electric and engine, and (6) absorption and engine. For each of these chiller combinations, all other combinations of cooling tower fan types, utility rates, plant part-load ratios (ratio of load to total chiller capacity), and ambient conditions given in Table 1 were considered, leading to 768 operating points (6 X 2 X 2 X 2 X 4 X 4) for evaluation of the control algorithm.
 Parameter Value

Design conditions

Wet-bulb temperature 80[degrees]F
Dry-bulb temperature 95[degrees]F

Electric chillers

Motor Fixed or variable-speed
Rated COP 6.0
Condenser-water flow rate 3 gpm/ton
Condenser-water pump power 15 W/gpm

Absorption chiller

Rated COP 1.0
Condenser-water flow rate 4 gpm/ton
Condenser-water pump power 15 W/gpm

Engine-driven chiller

Rated COP 1.5
Condenser-water flow rate 3 gpm/ton
Condenser-water pump power 15 W/gpm

Cooling tower

Motor Staged or variable-speed
Rated airflow 200 cfm/ton
Rated fan power 0.4 W/cfm

Utility costs

Electrical energy 0.05 or 0.15 $/kWh
Gas energy 0.40 or 1.20 $/therm

Plant operating conditions

Plant part-load ratio 0.25, 0.5, 0.75, 1.0

Wet-bulb temperature 50[degrees]F, 60[degrees]F, 70[degrees]F,
 80[degrees]F

Dry-bulb/wet-bulb difference 15[degrees]F


Figure 6 shows the effect of tower airflow and electric chiller part-load ratio on plant energy costs per ton of cooling provided for a plant that utilizes equally sized electric (variable-speed) and absorption chillers with an overall plant part-load ratio of 0.5 at the design ambient conditions ([T.sub.wb] = 80[degrees]F, [T.sub.db] = 95[degrees]F). Lines of constant plant costs are shown over the range of about 0.0.085 to 0.125 $/ton for the entire range of tower airflow (20% to 100% of the design tower airflow) and electric chiller part-load ratios (0.1 to 0.8).

[FIGURE 6 OMITTED]

Figure 6 indicates that the optimal tower airflow for this operating condition is about 50% of the design airflow and relatively independent of the relative chiller loadings. Greater airflow results in improved chiller performance but at the expense of increased fan power. Lower airflow results in lower fan power but with higher chiller energy input requirements. The optimum results from these trade-offs, which depends on the total plant load. The penalty associated with operating the cooling tower at the design airflow as compared with the optimal flow would be about 15% for these operating conditions if the electric chiller is heavily loaded. Although the penalty is much smaller if the absorption chiller loading is maximized, the optimal chiller policy for this case is to maximize loading on the electric chiller.

Benchmark Comparisons

The performance of the cooling tower control algorithm was evaluated using the simulation tool. Simulated plant costs for the simple control algorithm were compared with those for optimal tower airflow. The bechmark optimal results were determined by minimizing the plant costs of Equation 20 with respect to tower airflow using a one-dimensional golden-section search algorithm applied to the plant model. The comparisons were performed for the systems and operating conditions described in the previous section (Table 1) and assuming that the multiple chillers were loaded evenly. The comparisons were insensitive to whether the chillers were loaded evenly or not.

Figure 7 shows comparisons between the cooling plant costs for optimal and near-optimal control when applied to the hybrid, all-electric, and all-gas-driven chiller plants. Overall, the near-optimal control algorithm gives performance that is within 1% of the minimum power consumption. The method worked extremely well in all cases considered.

[FIGURE 7 OMITTED]

CONTROL STRATEGY IMPLEMENTATION

The cooling tower control algorithm involves determination of a relative tower airflow using Equations 16, 17, or 18. The different equations are for different fan types and give tower airflow relative to the maximum tower airflow if all tower fans were operating at the highest fan speed. As a result, the maximum relative airflow must be constrained to 1. There are additional constraints on the temperature of the supply water to the chiller condensers, which are necessary to avoid potential chiller maintenance problems. Some chillers have a low limit on the condenser water supply temperature that is necessary to avoid lubrication migration from the compressor. A high temperature limit is also necessary to avoid excessively high pressures within the condenser, which can lead to compressor surge. If the condenser water temperature falls below the low limit, then it is necessary to override the open-loop tower control and reduce the tower airflow to go above this limit. Similarly, if the high limit is exceeded, then the tower airflow should be increased as required.

At each decision interval (e.g., five minutes), the following steps can be executed to determine the setpoint for the relative cooling tower airflow:

1. Evaluate the time-averaged values of the condenser water supply temperature and overall plant cooling load over a fixed time interval (e.g., five minutes).

2. If the condenser water supply temperature is less than the low limit, then reduce the setpoint for the relative tower airflow by a fixed increment and exit the algorithm. Otherwise go to step 3. For fans with discrete settings, the setpoint increment should be chosen so that a single fan changes speed by a single step.

3. If the condenser water supply temperature is greater than the high limit, then increase the setpoint for the relative tower airflow by a fixed increment and exit the algorithm. Otherwise go to step 4.

4. If the chilled-water load has changed by a significant amount (e.g., 10%) since the last control change, then go to step 5. Otherwise exit the algorithm.

5. Use the current measured cooling load to estimate the PLR (load relative to rated capacity) for each chiller.

6. Use individual chiller part-load ratios, performance information at rating conditions for the chillers and cooling tower, and utility energy rates for the current rate period to determine a relative cooling tower airflow setpoint using Equation 16, 17, or 18.

The relative tower airflow must be converted to a specific set of tower fan settings. The total cooling tower airflow is approximately linear with individual tower cell fan speed and the number of fans operating, such that

[[gamma.sub.twr] = 1/[N.sub.twr] [[N.sub.twr].summation over (i - 1)] [[gamma].sub.twr,i], (22)

where[[gamma].sub.twr,i]is fan speed control function (0 to 1) for the ith cooling tower cell fan and [N.sub.twr] isthe number of cooling tower cells.

Equation 22 can be used to determine the required fan settings for a given total relative airflow. However, first it is necessary to have specific sequencing rules for the order in which fans should be turned on and off. As previously discussed, the best fan settings for a given airflow result from operating the maximum number of fans at the lowest possible speeds. In other words, the fans operating at the lowest speeds should be incremented first when adding fan capacity. For fans with discrete speed settings, this means that all tower fans should be operating at low speed before any fan speeds are increased to the next speed increment. Fan capacity should be removed in the reverse order in which it was added.

The process for converting from relative tower airflow setpoint to specific fan settings depends on the type of fan as described below:

1. Fans with Discrete Speed Settings. For multiple cells having fans with discrete speed settings (e.g., two-speed), Equation 22 and the sequencing rules can be used to construct a table relating airflow to fan-speed settings for individual cooling tower cells. The minimum tower airflow would be that associated with a single cell operating at its minimum fan setting (e.g., half-speed). The next increment of airflow would be associated with two cells operating at their minimum speed and so on. The conversion process between a relative airflow setpoint and the discrete fan control involves choosing the set of discrete fan settings from the table that produces a tower airflow closest to the desired flow, assuming that airflow is proportional to fan speed. In general, it is better to have greater rather than less than the optimal airflow. A good rule of thumb is to choose the set of discrete fan controls that results in a relative airflow that is closest to, but not more than 10% less than, the target relative airflow.

2. Variable-Speed Fans. If the cooling tower cells have only variable-speed fans, then all operating fans should be set at identical speeds. If possible, all tower cells should be operated unless the required fan speed would fall below a minimum allowable setting (e.g., 0.2). According to Equation 22, the fan-speed setting if all cells are operating is equal to the relative airflow setpoint. If the calculated setting for this case is below the minimum allowable setting, then the number of operating cells should be reduced by one and Equation 22 should be used to determine the required setting. This process should be repeated as necessary until the calculated fan setting is above the minimum allowable.

3. Discrete and Variable-Speed Fans. If the cooling tower cells have a mix of fans with variable-speed and discrete settings, then the goal is to operate the fans as close to the same speed as possible. For the discrete fans, Equation 22 and the sequencing rules can be used to construct a table relating airflow to fan-speed settings for individual cooling tower cells (see step 1). The first entry in the table should be for all discrete fans turned off. Then, for each entry in the table, the required fan control settings for the variable-speed fans that give the required relative airflow are determined using Equation 22, assuming that all the variable- speed fans operate at the same speed (see step 2). Then, the fan settings are selected from the table using the criteria that the variable-speed setting should be closest to, but less than, the maximum discrete fan setting.

COOLING TOWER CONTROL EXAMPLE

Consider an example plant consisting of one variable-speed electric and one engine-driven chiller, each having a rated cooling capacity of 500 tons. The rated COP for the electric chiller is 6, resulting in a rated electrical power input requirement of

[E.sub.ch,rated,E-1] = [Q.sub.ch,rated,E-1]/[COP.sub.rated,E-1] = 500 tons/6 . 3.517 kW/ton = 293.1 kW .

The rated COP for the engine-driven chiller is 1.5. Then, the rated requirement for energy input to the gas chiller is

[E.sub.ch,rated,G-1] = [Q.sub.ch,rated,E-1]/[COP.sub.rated,E-1] = 500 tons/1.5 . 3.517 kW/ton = 1172 kW .

The cooling tower has two cells, each having two-speed fans. The tower design approach range from manufacturers' data are 7[degrees]F and 10[degrees]F. The rated airflow and fan power 200 cfm/ton and 0.4 W/cfm. Therefore, the fan power for the total rated plant cooling capacity is

[P.sub.twr,rated] = 200 cfm/ton . 0.0004 kW/cfm . 1000 tons = 80 kW .

596 HVAC&R RESEARCH

From energy balances on the chillers operating at rated capacities and COPs, the fraction of the cooling tower heat rejection associated with each chiller is:

[f.sub.[twr,rated,E-1]] = [Q.sub.[ch,rated,E-1]](1 + [1/[COP.sub.[rated,E-1]]])/[Q.sub.[ch,rated,E-1]](1 + [1/[COP.sub.[rated,E-1]]]) + [Q.sub.[ch,rated,G-1]](1 + [1/[COP.sub.[rated,G-1]]]) = 500(1 + 1/6)/500(1 + 1/6) + 500(1 + 1/1.5) = 0.412

[f.sub.[twr,rated,G-1]] = 1 - [f.sub.[twr,rated,E-1]] = 0.588

From the characteristics given in Figures 2 and 4, it is found that the rated sensitivity factor for chiller input energy with respect to changes in condenser water temperature is about 0.015[degrees][F.sub.-1] for both chiller types. This is a relatively typical value for chillers.

Consider a time during the on-peak period where the total plant load is 700 tons, electrical energy costs are $0.1/kWh, and gas energy costs are $0.80/therm. If the chillers are equally loaded, then both chillers have a part-load ratio of 0.7 under these conditions. For two-speed fans, the relative tower airflow is then determined using Equation 17 so that

[[gamma].sub.twr] = [{1/2([a.sub.[twr,rated]] + [r.sub.[twr,rated]])([N.sub.ch].summation over (i=1)]([PLR.sub.[ch,i]][f.sub.[twr,rated,i]]))[[N.sub.ch].summation over (i=1)]([PLR.sub.[ch,i,rated,i]] [[ER.sub.i][E.sub.[ch,rated,i]]]/[[ER.sub.e][P.sub.[twr,rated]]])}.sup.[1/3]] = [{1/2(7F + 10F])(0.7 x 0.411 + 0.7 x 0.588)}.sup.[1/3]] x [{[0.7 x 0.015]/[0.1 $/kWh x 80 kW](0.1 $/kWh x 293.1 kW + 0.80 $/therm x 0.03412 therm/kWh x 1172 kW)}.sup.[1/3]] = 0.784 .

At this load, the cooling tower fans should operate at approximately 78% of the maximum tower airflow. In order to convert [[gamma].sub.twr] into a specific tower control, it is necessary to define the tower sequencing. Table 2 gives this information in a form that specifies the relationship between [[gamma].sub.twr] and tower control for this example.

For a specific chilled-water load, the fan control should be the sequence of tower fan settings from Table 2 that results in a value of [[gamma].sub.twr] that is closest to, but not more than, 10% less than the value computed with Equation 17. For this example, sequence number 3 would represent the best choice.
Sequence Number [[gamma].sub.[twr]] Tower Fan Speeds

 Cell #1 Cell #2

1 0.25 Low Off
2 0.50 Low Low
3 0.75 High Low
4 1.00 High High


CONCLUSIONS

The algorithm developed in this paper is similar in nature to the approach developed by Braun and Diderrich (1990) for all-electric plants that appears in the 2003 ASHRAE Handbook--HVAC Applications (ASHRAE 2003). However, the current algorithm is general for cooling plants that incorporate any combination of electric and gas-driven chillers. The control algorithm determines cooling tower fan settings in response to loadings on individual chillers. Parameters of the algorithm are evaluated using design information for the chillers and cooling tower fans. In addition to reducing operating costs, use of the open-loop control strategy simplifies the control and improves the stability of the tower control compared with the use of a constant condenser water supply or approach to wet-bulb.

ACKNOWLEDGMENTS

The financial support of ASHRAE under RP-1200 and the technical support provided by the Project Monitoring Subcommittee that included Jay Kohler, Paul Sarkisian, and Dharam Punwani are greatly appreciated.

NOMENCLATURE

[a.sub.twr] = cooling tower approach to wet-bulb temperature ([T.sub.cws]--[T.sub.wb])

[C.sub.ch] = instantaneous cost of operating chillers

[C.sub.twr] = instantaneous cost of operating cooling tower

[COP.sub.i] = coefficient of performance for the ith chiller

[E.sub.ch,i] = rate of energy input for ith chiller (gas or electric)

[ER.sub.e] = cost per unit of electrical energy

[ER.sub.i] = cost per unit energy input (heat or electricity) for the ith chiller (gas or electric)

[f.sub.twr,i] = fraction of cooling tower heat rejection associated with ith chiller

[N.sub.ch] = total number of chillers (gas and electric)

[N.sub.twr] = number of cooling tower cells

[P.sub.cwp] = total condenser water pump power

[P.sub.twr] = total cooling tower fan power

[PLF.sub.i] = part-load factor for ith chiller defined as the ratio of energy usage to the value at the rating condition

[PLR.sub.[ch,i]] = part-load ratio for ith chiller (gas or electric)--load relative to rated chiller capacity

[PLR.sub.plt] = part-load ratio, plant--total cooling load relative to rated cooling capacity for plant

[Q.sub.ch,i] = cooling load for ith chiller

[Q.sub.twr] = cooling tower heat rejection rate

[r.sub.twr] = cooling tower range ([T.sub.cwr]--[T.sub.cws])

[S.sub.ch,i] = sensitivity of ith chiller input energyto changes in condenser water temperature

[T.sub.cwr] = condenser water return temperature

[T.sub.cws] = condenser water supply temperature

[T.sub.db] = ambient dry-bulb temperature

[T.sub.wp] = ambient wet-bulb temperature

[[gamma].sub.[ch,i]] = control function that determines whether the ith chiller (gas or electric) is on or off (0 or 1)

[[gamma].sub.twr] = control function that specifies the relative airflow for the cooling tower

[[gamma].sub.[twr,i]] = control function that specifies the relative fan speed or airflow for an individual cooling tower cell

Additional Subscript

rated = evaluated at rating conditions

REFERENCES

ASHRAE. 2003. 2003 ASHRAE Handbook--HVAC Applications, Chapter 41, "Supervisory Control Strategies and Optimization," pp. 41.1--41.39. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Braun, J.E. 1988. Methodologies for the design and control of central cooling plants. PhD dissertation, University of Wisconsin--Madison.

Braun, J.E., S.A. Klein, J.W. Mitchell, and W.A. Beckman. 1989a. Applications of optimal control to chilled-water systems without storage. ASHRAE Transactions 95(1):663--75.

Braun, J.E., S.A. Klein, J.W. Mitchell, and W.A. Beckman. 1989b. Methodologies for optimal control of chilled-water systems without storage. ASHRAE Transactions 95(1):652--62.

Braun, J.E., S.A. Klein, and J.W. Mitchell. 1989c. Effectiveness models for cooling towers and cooling coils. ASHRAE Transactions 95(2):164--74.

Braun, J.E., and G.T. Diderrich. 1990. Near-optimal control of cooling towers for chilled-water systems. ASHRAE Transactions 96(2):806--13.

Braun, J.E. 2006. Optimized operation of chiller equipment in hybrid machinery rooms and associated operating and control strategies, ASHRAE RP-1200 final report. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Hackner, R.J., J.W. Mitchell, and W.A. Beckman. 1985. HVAC system dynamics and energy use in buildings--Part II (RP-321). ASHRAE Transactions 91(1B):781--95.

Klein, S.A., D.R. Nugent, and W.A. Beckman. 1988. Investigation of control alternatives for a steam turbine driven chiller. ASHRAE Transactions 94(1):627--43.

Koeppel, E.A., J.W. Mitchell, S.A. Klein, and B.A. Flake. 1995. Optimal supervisory control of an absorption chiller system. HVAC&R Research 1(4):325--42.

Lau, A.S., W.A. Beckman, and J.W. Mitchell. 1985. Development of computer control--Routines for a large chilled-water plant. ASHRAE Transactions 91(1B):766--80.

Sud, I. 1984. Control strategies for minimum energy usage (RP-253). ASHRAE Transactions 90(2A):247--77.

James E.Braun, PhD

Fellow ASHRAE

James E. Braun is a professor of mechanical engineering, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN.(ASHRAE 2003). However, this strategy is not appropriate for hybrid plants because it is based on minimizing input energy usage rather than cost.
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Author:Braun, James E.
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Date:Jul 1, 2007
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