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Methodology for determining the optimal operating strategies for a chilled-water-storage system--part I: theoretical model.

Introduction

Thermal energy storage (TES) is the concept of generating and storing energy in the form of heat or cold for future use. This concept has been used for centuries, but only recently have large electrical users taken advantage of this technique for demand-side management and cost reduction. This article focuses on a chilled-water (ChW) system with a naturally stratified ChW-storage tank, which is a subclass of TES systems. It uses the principle that warm return water and cool stored water tend to stratify due to the density differences to keep the water from mixing.

The advantages of a TES system are summarized by the following three concepts. Except for capital cost savings due to reduced equipment size, the TES system is designed to avoid high electric utility energy or demand charges by shifting cooling production from the time of high electrical demand and energy costs to the time of low costs. This is the general design purpose of a ChW TES system, and most systems are operated following this strategy. Second, a TES system provides an opportunity to decouple the production and the consumption of the ChW This decoupling effect could be utilized to provide increased flexibility, reliability, or backup capacities for the control and operation of the system (ASHRAE 2003a). The third concept is that the plant performance can also be improved by loading chillers at the optimal part-load ratio (PLA) or at night when the ambient wet-bulb (WB) temperature is low. Sometimes, the plant efficiency can be improved further by shifting the cooling load to more efficient chillers (such as new electric centrifugal chillers), thus avoiding the operation of less efficient chillers. Documented examples include ChW-storage installations that reduce annual energy consumption on a kWh basis for air conditioning by up to 12% (Bahnfleth and Joyce 1994). Consequently, TES is not only cost effective, but it also could be energy effective if operated properly.

Much of the success experienced by TES technology in the past can be attributed to electric demand charges and to capital cost incentives offered through utility rebate programs. Concerns are that TES is not a green technology and that changes in the power production industry may eliminate both demand charges and rebates. However, using an example facility, Caldwell and Bahnfleth (1997) found that without electric rebate incentives or rebates, stratified ChW TES yielded a first cost savings of 9% to 17% and a life-cycle savings of 33% to 36% over two non-TES plant alternatives for the example facility. It was concluded that stratified ChW TES was a viable technology, even without the presence of electric rebate incentives or rebates, and that it was a sustainable technology for the foreseeable future.

In practice, many ChW-storage systems are manually operated based on operators' experiences targeting some simple objectives, such as avoiding chillers running during the on-peak hours, charging the tank as soon as possible during the off-peak hours, and fully loading the chillers for the initial loading period. During the winter months when utility rate structures change, many ChW-storage tanks are not in use. Such kinds of operations may reap part of the benefits from the thermal storage tank, but they cannot make full use of the advantages of a TES system.

There are many reasons why the TES capabilities are not fully exploited. One reason could be that sophisticated controllers and adequate controls sensors are not available. The fear of prematurely depleting the tank during the on-peak hours also forces the operators to take a more conservative attitude in tank operations. The various operation modes together with complicated rate structures also enhance the difficulties and complexities of determining the optimal operating strategies.

Differences between water storage and ice storage

A basic review of the current TES studies shows that ice storage has become the most prevalent (ASHRAE 2006). However, large-scale applications (over 10,000 ton-h [35,169 kW-h]) are dominated by the use of ChW-storage systems (Andrepont 2006).

Although they share the same idea of shifting electrical load, there are still obvious differences between ChW and ice-storage systems. Compared to an ice-storage system, the primary advantage of a ChW-storage system is that the produced ChW temperature can be the same (39[degrees]F-42[degrees]F[3.9[degrees]C5.6[degrees]C] normally) when the system shifts between charging and discharging modes. Standard commercial chillers can be used, and the control is much easier. A ChW-storage system has no efficiency penalty since it is not necessary to produce extremely low temperature water (22[degrees]F-26[degrees]F [-5.6[degrees]C to -3.3[degrees]C] normally). There will be more capacity loss, however, due to mixing effects and heat losses through the tank wall. A second difference is that the charging and discharging rate of a ChW storage system is determined by the acceptable ChW flow rate and stored warm and cool water temperatures all the time. The heat transfer rate for an ice-storage system is limited by the heat exchanger area, secondary fluid flow rate and inlet temperature, and the thickness of ice at any time. Complicated correlations are required to calculate the charging and discharging effectiveness (Drees and Braun 1995). In addition, the actual inventory of the water tank is dependent upon the temperature difference between the tank inlet and outlet while the capacity of the ice tank is determined by the ice volume.

The experiences on an ice-storage system may be used as a good reference for a water-storage system. However, the comparison shown above indicates that the modeling of a ChW-storage system and an ice-storage system has obvious differences. Such differences may lead to inconsistent simulation and optimization conclusions on the operations of two systems. Therefore, it is not prudent to transfer the conclusions from an ice-storage system to a ChW-storage system without a thorough analysis.

Tank performance index

Several metrics have been used to quantitatively describe the performance of a ChW tank. The cycle thermal efficiency of a stratified tank is the ratio of the integrated discharge capacity for a complete discharge process to the integrated charge capacity for the preceding complete charge process for a true cycle in which initial and final states of the tank are identical (Wildin and Truman 1985). This index is useful as a measure of long-term tank performance because the small differences between the initial and final states of water in the tank become insignificant compared to these much larger capacities. However, it measures only capacity losses through the tank wall and does not account for mixing internal to the tank. Tran et al. (1989) tested several large ChW-storage systems and proposed a figure-of-merit (FOM) to reflect the loss of usable capacity. The FOM is the ratio of integrated discharge capacity for a given volume to the ideal capacity that could have been withdrawn in the absence of mixing and losses to the environment. The FOM may be difficult to measure in the field because many operating ChW-storage sites cannot conduct full-cycle tests running for 24 h or longer. A half-cycle FOM has been defined as the ratio of integrated charge or discharge capacity to the theoretical capacity contained in one tank volume (Bahnfleth and Musser 1998). It measures capacity lost to mixing in a half-cycle (single charge or discharge process) rather than a full cycle. A lost capacity in a charge process was defined as the capacity that could not be stored because the system could not continue to cool water as it approached the inlet temperature (Bahnfleth and Musser 1998). It is defined relative to an application-specific limiting temperature. Capacity is "lost" in a discharge half-cycle when water in the tank exits at a temperature above the upper limit that can be utilized by the process served.

A mixing effect leads to a reduction of usable ChW volume, while a heat loss to the environment results in an increase of the ChW bulk temperature. It is necessary to treat these two kinds of capacity losses in different ways for a ChW storage system.

TES studies

An in-depth literature search and study shows that most research is focused on ice-storage systems since it is the most popular TES system. The studies on ChW-storage systems are mainly concentrated on field experiment testing and numerical simulations of the tank performance.

Tran et al. (1989) tested six ChW-storage systems and found that well-designed storage tanks had an FOM of 90% or higher for daily complete charge and discharge cycles and between 80% and 90% for partial charge and discharge cycles. Bahnfleth and Musser (1998) found that the lost capacity was roughly 2% of the theoretical capacity available when a minimum outlet temperature limit was applied, while as much as 6% could be lost for discharge processes performed at the same flow rate for typical limiting temperatures. Discharge cycle lost capacity was significantly decreased by reducing the inlet flow rate. In a dynamic mode of operation, the effects of mixing overtook the influence of other parameters, but the effect of wall materials could not be neglected when the tank was in an idle status (Nelson et al. 1999b). Caldwell and Bahnfleth (1998) found that mixing was localized near the inlet diffuser and directly related to flow rate. Nelson et al. (1999a) proposed the definition of the mixing coefficient, which was expressed as a function of Reynolds number (Re) and Richardson number (Ri).

Some researchers developed dynamic or static simulation models to study the thermal performance of a stratified ChW-storage tank. Gretarsson et al. (1994) derived a fundamental energy balance model based on a one-dimensional plug-type flow approach. Studies showed that the thermocline thickness could be 3% to 7% of the water height. Homan et al. (1996) grouped the capacity loss into heat transfer through the tank walls, conduction across the thermocline, and the flow dynamics of the charge and discharge process, and they found that the flow dynamics were generally orders of magnitude more important than the other factors. Published data showed current storage tanks generally operated at efficiencies of 50% to 80%.

This research indicates that considerable capacity loss may occur when a maximum tank outlet temperature limit is applied, especially during a discharge cycle at a higher flow rate. The tank discharge rate should be controlled to minimize the mixing effect near the inlet diffuser. These findings could place some constraints on the optimization of the TES system and also provide insights to simply quantify the tank performance.

The TES control strategies are classified as conventional and nonconventional. Conventional tank control strategies include full storage and partial storage. The partial storage can be further divided into chiller-priority and storage-priority. Demand limiting control or load-limiting control may be combined with any of the above control strategies (ASHRAE 2003b). These strategies are often used as benchmarks compared with nonconventional control strategies. Forecasts of building cooling requirements and weather conditions are not required for chiller priority control but are required for other strategies.

The nonconventional control strategies include optimal strategies and near-optimal strategies or rule-based control strategies. An optimal TES control strategy is a sequence of plant flow rate operations that can minimize the operating cost of the targeted system region. It is a complicated function of several factors, such as utility rates, load profiles, plant characteristics, tank performance, loop characteristics, and weather. Dynamic programming or some direct search methods can be used to find the globally optimal solution in a reasonable amount of time. The optimal results from dynamic programming may maximize the savings. However, in most cases, such an optimal sequence is difficult to follow since there is no clear control logic inside. Some researchers developed heuristics by studying the optimal trajectories and summarizing them into some rule-based control strategies or so-called near-optimal controls. They consist of different conventional control strategies with some judgment clauses.

TES control can be divided into charging and discharging strategies. Most systems share the same charging strategies. Charging should be initiated when the building load is lower and off-peak electrical rates are in effect. The discharging strategy could be different for various systems when different control strategies are adopted. Chiller-priority control operates the chiller, up to its available capacity, to meet loads. It is the most simple and most commonly applied with the chiller in series upstream of storage, but it minimizes the load shifted by the TES system and works well economically when the utility rate does not include demand charges or time-of-use (TOU) electricity rates (Henze 2003b). Storage-priority control meets the load from storage up to its available discharge rate. This allows for maximal load shifting but comes with the risk of depleting the storage capacity prematurely by under-predicting loads (Henze 2003b). Load forecasting is required to maximize its benefits (Wei et al. 2002). Simpler storage-priority sequences using constant discharge rates, predetermined discharge rate schedules, or pseudo-predictive methods have also been used (ASHRAE 2003a). Full-storage control strategy only applies when the tank capacity is large enough to ensure running a chiller during the on-peak period is not necessary. It could be regarded as a special case for the storage-priority strategy. These conventional control strategies are easy to follow and can reap part of the cost-saving benefits. They will be used as a benchmark when it comes to calculating the savings potential of new strategies.

TES optimization is finding a combination of different control strategies during specific periods to achieve minimal energy consumption or demand cost over a utility billing period. It can be divided into component-based optimization and system-based optimization. For component-based optimization, each component is represented as a separate subroutine with its own parameters, controls, inputs, and outputs. The models used by Henze et al. (1997b) predicted cooling plant and distribution system power with a component-based simulation that is appropriate for simulation studies. Alternatively, for system-based optimization, plant and distribution system power can be simplified with empirical correlations, such as Braun (2007a). Drees (1994) used curve-fits of plant power consumption in terms of cooling load and ambient WB temperature.

Most references are related to an ice-storage system, but several ChW storage cases can still be found. Braun (1992) described a comparison of control strategies for a partial ice-storage system installed in an office building located in Milwaukee, Wisconsin. The results indicated that the load-limiting strategy provides a near-optimal control in terms of demand costs for all environmental conditions considered. Dorgan and Elleson (1993) used the term operating strategy to refer to full-storage and partial-storage operation. That discussion focuses on design-day operation and does not discuss operation under all conditions. Krarti et al. (1995) evaluated chiller-priority and storage-priority control strategies for ice systems as compared with optimal control for a wide range of systems, utility rate structures, and operating conditions. Similar to Braun (1992), they concluded that load-limiting, storage-priority control provided near-optimal performance when there were significant differentials between on-peak and off-peak energy and demand charges. However, without TOU energy charges, chiller-priority control did provide good performance for individual days when the daily peak power was less than the monthly peak.

Drees and Braun (1996) found that for ice storage, a simple and near-optimal approach was to set target demand cost to zero at the beginning of each billing period. The optimization results were used to develop a rule-based discharge strategy. Henze et al. (1997b) developed a simulation environment that determined the optimal control strategy to minimize operating cost, including energy and demand charges, over the billing period. The simulation tool was used to compare the performance of chiller-priority, constant-proportion, storage-priority, and optimal control. Henze et al.(1997a) presented a predictive optimal controller for use with real-time-pricing (RTP) structures. The controller calculates the optimal control trajectory at each time step (e.g., 30 min), executes the first step of that trajectory, and then repeats that process at the next time step. The efficiencies of the cooling plant in the ChW model and ice-making model are assumed constant. The component-based plant optimization is described in detail by Krarti et al. (1995). The state of the ice-storage tank is defined by state-of-charge and rate-of-change variables with constraints. Dynamic programming is used to find the optimal control trajectory.

Hajiah (2000) investigated the effects of using both of building thermal capacitance and an ice-storage system to reduce total operating costs of a central cooling plant while maintaining adequate occupant comfort conditions in buildings. An optimal controller of a central cooling plant was developed using the results from a simulation environment. It was implemented and tested. A neural-network-based optimal controller was developed by Massie (2002) to control a commercial ice-storage system for the least cost under an RTP rate. It is robust in finding solutions given any price structure, building cooling load, and equipment operating conditions. Because of its ability to learn patterns, it self-calibrates to equipment operating characteristics and does not require an expert to fine tune.

Henze and Schoenmann (2003) presented a model-free reinforcement learning controller for optimal operation of TES systems. The reinforcement learning controller learned to charge and discharge a thermal storage tank based on the feedback it received from past control actions. The performance of this controller was evaluated by simulations, and the result showed that it had strong capability to learn a difficult task of controlling TES with good performance. However, cost savings were less when using a predictive optimal controller.

Henze (2003a) investigated whether thermal storage systems could be controlled effectively in situations where cooling loads, noncooling electrical loads, weather information, as well as the cost of electricity were uncertain and had to be predicted. The analysis shows that the reduction in achievable utility cost savings is small when relying on RTP electricity rates that are made available by the utility only 1 h ahead instead of an entire day ahead. Consequently, uncertain electrical utility rates do not imperil the superior cost-saving benefits of cool storage when governed by predictive optimal control.

A module for ice-based TES systems was developed and integrated within EnergyPlus by Ihm et al. (2004). The TES module uses building load and system thermodynamics (BLAST) models for two direct ice systems (ice-on-coil external melt and ice harvester) and one indirect ice system (ice-on-coil internal melt). It provides designers and facility operators with an effective simulation environment to determine the best control strategy.

A near-optimal control method was developed for charging and discharging of cool storage systems when RTP electric rates were available (Braun 2007a). The model includes a correlation for plant cooling capacity as a function of chiller supply temperature and ambient WB temperature and a correlation for plant power consumption as a function of chiller cooling load, chiller supply temperature, and ambient WB temperature from simulations that incorporated individual equipment models. A model was developed for the time dependence of typical RTP rates that depends on time of day and maximum temperature for the day. For charging of storage, it was found that a very simple, near-optimal strategy is to fully recharge storage with the chiller operating at maximum capacity during a period defined by when the RTP rates are lowest and the building is unoccupied. For discharging of storage, it was found that the best strategy is to use a storage priority control that maximizes the discharge rate of storage during a period defined by when RTP rates are highest, the building is occupied, and it is economical to utilize storage. For all other times, it is best to use chiller-priority control that minimizes the discharge rate of storage. The simplified method worked well in all cases and gave annual costs within approximately 2% of the minimum possible costs associated with optimal control.

Braun (2007b) evaluated the operating cost savings associated with employing the strategy developed by Braun (2007a) as compared with using chiller-priority control. In addition, operating cost savings associated with employing ice storage in combination with RTP rates were evaluated for both the near-optimal and chiller-priority strategies. For a range of systems employing ice storage with RTP rates, the cost savings associated with the near-optimal strategy compared to chiller-priority control were found to be as high as 60%, with typical savings between 25% and 30%. These savings are much more significant than savings associated with employing near-optimal control for cool storage systems when typical TOU utility rates are employed with demand charges. A similar level of savings was determined when comparing costs for the near-optimal control strategy applied to ice-storage systems with costs for systems not employing cool storage. However, relatively small savings were determined for use of ice storage when chiller-priority control is utilized. In many situations, the use of storage with chiller-priority control can actually result in higher costs than without storage. It can be concluded that chiller-priority control should not be employed in combination with RTP rate structures for cool storage systems. With conventional rates, the largest part of the cost savings opportunity is associated with reduced demand due to downsizing of the peak chiller cooling capacity. For application of cool storage with RTP utility rates, the opportunity for cost savings is much more sensitive to the control strategy employed.

Henze et al. (2008) described the investigation of the economic and qualitative benefits of adding a ChW-TES system to a group of large buildings in the pharmaceutical industry in southern Germany. From their findings, it can be expected to provide economic benefits as measured in energy cost savings, as well as qualitative merits such as the avoidance of numerous safety measures necessary for a ChW plant without storage (e.g., always operating at least two chillers) and a cost-effective addition of supplemental ChW plant cooling capacity. Moreover, the overall system reliability and availability will be significantly improved through the addition of a TES system. The near-optimal heuristics suitable for implementation in the actual pharmaceutical buildings is an on-going task.

Based on the reviews above, the current research can be summarized as follows. (1) Most TES operating and control researchers emphasize an ice-storage system or combining ice-storage active storage with building passive thermal storage. The study on a water-storage system operation is not sufficient. (2) Current studies on TES systems are on a case-by-case basis, and there is not a general method to find the optimal operating strategy. (3) Dynamic programming is used to obtain the optimal control strategies. Then, the near-optimal strategies are induced from the optimal trajectories. Such sophisticated routines are not easy to follow. It is also difficult to induce some logic from the optimal control strategies. (4) Studies on utilizing TES to enhance ChW plant performance are rare. The TES is regarded as more of a cost-management tool than an efficiency enhancement tool. (5) All studies used the ton and ton-h to depict the tank state change and tank inventory (Henze et al. 2008). It is acceptable for an ice-storage tank because the tank inventory is not affected by water temperatures but by the ice volume in the tank. However, for a water-storage tank, the return temperature may fluctuate a lot diurnally or seasonally. Consequently, such a description method will lead to inconsistent results.

The major objective of this article is to propose a generic methodology to determine cost-effective and reliable operating strategies for a ChW-storage system under a TOU rate structure. It combines TES optimization and plant optimization together and captures the main characteristics of the plant, TES tank, loop, load profiles, and utility rate structures. It is able to be generalized to some popular systems and find the optimal operating strategies quickly. In addition, the safety consideration is included and is adjustable to accommodate a conservative or an aggressive operating attitude. Except for the introduction of terms in the text, nomenclature can be found at the end of the article for identification of abbreviations, Greek symbols, and subscripts.

Methodology

Basic system configuration

Figure 1 shows the basic configuration of a naturally stratified ChW-storage system. A primary-secondary pump system is designed with variable-speed secondary pumps (SPMPs) and constant-speed primary pumps (PPMPs). All air-handling and terminal units are controlled by two-way control valves. The TES tank parallels the chillers and functions like a bypass with an extremely large volume. A pressure-sustaining valve (PSV) and a check valve are necessary to avoid a vacuum in the pipes if systems are above the water level. If the elevation of users is much higher than the tank water level, heat exchangers will be designed to transfer the cooling from the tank loop side to the user loop side. Under this situation, a temperature penalty will be applied. This system configuration is the most popular because it is easy to control. In retrofit projects, such a configuration is often adopted since the least system changes have to be made. As a result, this study will focus on this configuration. It is noted that this is only an illustration configuration and is not intended to show all system piping and control requirements in practice.

[FIGURE 1 OMITTED]

There are no modulating devices on the tank in this configuration. The tank charging or discharging flow rate is the difference between the plant-side total flow and the loop-side total flow. Since the loop-side flow rate cannot be controlled by the plant, the TES charging or discharging flow rate is determined by plant total ChW flow rate. The TES operation profile is, in fact, a profile of ChW total flow rate supplied by the plant. The plant total flow rate is also constrained by some limits, such as chiller evaporator maximum (avoiding erosion) and minimum (avoiding freezing) flow rates, PPMP maximum flow rate, and tank design maximum charge and discharge flow rate to avoid intense mixing.

Rate structure

The electric utility rate schedule is the main driving force for TES applications. Table 1 is a typical TOU energy and demand rate structure for a TES system. It is noted that there is a possibility that the definition of on-peak or off-peak hours for the energy rate is different from that for demand rate. For a specific control strategy, it is necessary to define an on-peak period and an off-peak period for one cycle, normally 24 h. For summer billing months and winter billing months, such a definition could be different when the electrical rate structure changes. In most cases, this definition matches the definition of on-peak and off-peak hours for energy or demand rates.

Objective function

For a typical ChW-storage system, the instantaneous electrical power consists of the following two components:

[P.sub.sys] = [P.sub.Plant] + [P.sub.non-plant], (1)

where [P.sub.sys] is the total power billed by the utility company, and [P.sub.non-plant] covers all other electricity usages excluding ChW production and distribution in the facilities, such as air-handling units (AHUs), terminal boxes, elevators, lighting, office equipment, etc. Models are needed to simulate the second term if it is covered in the bill. [P.sub.Plant] is the ChW production-related electricity consumption in the plant, and it is sum of the following items:

[P.sub.Plant] = [P.sub.CT] + [P.sub.CWP] + [P.sub.CHLR] + [P.sub.PPMP] + [P.sub.SPMP]. (2)

Typically, in a well-maintained chiller plant, more than half of the plant electricity consumption is attributed to chillers, while the other half is split between pumps and fans. Miscellaneous power attributed to plant lighting and plug loads is considered to be negligible compared to the major plant loads.

In most cases, the optimization target of TES system operation is to minimize the operating cost within a billing period, such as a year. Most utilities have only two distinct rate periods, known as on-peak hours and off-peak hours. For each month, the objective function is the monthly cost function, which can be stated as (Krarti et al. 1995)

[C.sub.i] = [2.summation over (v=1)] [R.sub.d,i,v][P.sub.i,v] + [[n.sub.i].summation over (i=1)][24.summation over (k=1)] [R.sub.e,i,k][P.sub.k] [DELTA]t, (3)

where [R.sub.d,i,v] is the demand charge rate for rate period v and month i, [P.sub.i,v] is the billed demand kW in period v and month i, [R.sub.e,i,k] is the energy charge rate at hour k and month i, [P.sub.k] is the total power incurred from the system at hour k, [N.sub.i] is the days in month i, and [DELTA]t is a unit time step of 1 h. The calculation of [P.sub.i,v] could be complicated when a ratchet is defined. The demand and energy charge rates are fixed when a contract is signed.

Optimal operating strategy search method

Figure 2 shows the flowchart of the search procedure for the optimal control strategy. For each month, a search is performed to find a feasible and most cost-effective TES operating strategy. Then, a plant optimization program is launched to find the optimal controlled variables for this month.

The operating strategy can be described with a control strategy and the maximal numbers of chillers that can be staged-on during the off-peak and on-peak periods. This number should be no less than zero and no higher than the number of installed chillers in the plant. The limitation on the number of chillers running is a kind of demand limiting because, for a multi-chiller plant, the ChW-related power is directly proportional to the number of chillers running. Each control strategy consists of a series of control logic, which is used to calculate the plant total ChW flow rate and the number of chillers staged-on for each time step.

[FIGURE 2 OMITTED]

An optimal TES control strategy is a trade-off of benefits and risks. The benefit is billing cost savings, and the risk is the potential of depleting the tank prematurely, which forces operators to run additional chillers during the on-peak hours. In the simulation, the uncontrolled variables, such as loop total cooling load, loop delta-T, and weather conditions, are assumed to be perfectly known. However, in controller design, these variables will be different from those in simulations. To ensure the selected control strategy is reliable, a minimum tank level set-point is defined to filter all the combinations. The higher the minimum level set-point is, the lower the risk of the strategy is.

Within the search loop, all combinations of available control strategies and the maximal chiller number during the off-peak and on-peak periods are explored. The hourly tank water level and system total power are simulated with a model called a system model. A minimum tank level set-point is predefined to prevent premature depletion. The minimal water level in the current month is compared with the set-point to determine if the current combination is acceptable. For all acceptable combinations, the scenario with the lowest monthly billing cost will be chosen as the optimal operating strategy for the current month.

Plant optimization

A plant optimization procedure will be performed right after the TES control strategy optimization procedure. The variables that could be optimized include, but are not limited to, the chiller ChW leaving temperature, condenser water (CW) flow rate of each chiller, and cooling tower (CT) approach temperature. Some constraints are applied to these variables, such as the minimal tank water level, the lowest ChW leaving temperature the chiller can produce, and the highest ChW supply temperature the loop can tolerate.

Figure 3 shows the general physical configuration of a ChW system. All the variables shown are set-points to be optimized. In practice, these set-points are maintained by adjusting the equipment speed or control valve position with a proportional-integral-differential (PID) controller. As mentioned before, except for continuous control variables, discrete control variables will also need to be optimized, such as sequencing of chillers, CTs, and pumps. The constraints on the equipment operation, such as maximum and minimum flow rates, limit the possible number of combinations of control variables.

[FIGURE 3 OMITTED]

This is a nonlinear programming (NLP) problem, and it can be solved with the GRG (generalized reduced gradient) nonlinear solver in the standard Excel Solver. This method and specific implementation have been proven in use over many years as one of the most robust and reliable approaches to solving difficult NLP problems.

System power simulation

The flowchart of a system model is shown in Figure 4. It is used to calculate the tank water level at the end of time step and the system total power for each time step. This model includes six sub-models, and each of them will be introduced in the following sections.

The advantage of such a system model is that each sub-model is independent, and its function is explicitly specified. It also clearly describes the relationships among plant, loop, and TES tank. For different applications, the user may replace them with self-built sub-models or make minor changes on the original ones. In addition, the user can design a new control strategy to maximize the savings based on case-by-case considerations.

TES tank modeling

In this study, the tank ChW volume ratio and the tank charging or discharging flow rate are utilized to describe the tank state and inventory change rate. In this context, the state-of-charge x is explained as the ChW volume ratio in the tank. The state of a full tank is unity and of an empty one is zero. The primary controlled variable [V.sub.Tank,ChW] in gallons per minute (GPM) is defined as the rate change of the state-of-charge [x.sub.k]:

x = [U.sub.Tank,ChW]([T.sub.ChW][less than or equal to][T.sub.ref]) / [U.sub.Tank], (4)

[V.sub.Tank,ChW] = [phi] [dU.sub.Tank,ChW] / dt = [phi] ([V.sub.Plant,ChW] - [V.sub.Lp,ChW]), (5)

[x.sub.k+1] = [x.sub.k] + [phi] ([V.sub.Plant,ChW,k] - [V.sub.Lp,ChW,k]) x [DELTA]t / [U.sub.Tank], (6)

subject to the constraints:

[x.sub.min] [less than or equal to] [x.sub.k] [less than or equal to] [x.sub.max],

0 [less than or equal to] [V.sub.Plant,ChW,k] [less than or equal to] [V.sub.Plant,ChW,max],

[V.sub.Tank,ChW,min] [less than or equal to] [V.sub.Tank,ChW,k] [less than or equal to] [V.sub.Tank,ChW,max].

[FIGURE 4 OMITTED]

The chiller ChW leaving temperature is fixed in one cycling period and could be adjusted month by month. The plant ChW leaving temperature is normally different from the loop ChW supply temperature. The following assumption is made on the loop supply temperature:

[T.sub.Lp,ChW,S] = [T.sub.Plant,ChW,S] + [DELTA][T.sub.S], (7)

where [DELTA][T.sub.S] is the ChW temperature rise due to pumping, piping heat losses, and tank heat losses. It is around 0.5[degrees]F-1.5[degrees]F(0.3[degrees]C-0.8[degrees]C), depending on the system characteristics, such as loop differential pressure (DP), piping and tank insulations, and pump efficiencies.

The loop return water temperature may fluctuate diurnally, low at night and high in the daytime. As the return temperature is close to the ambient temperature and the PPMP head is much smaller than SPMP head, the temperature rise can be neglected:

[T.sub.Plant,ChW,R] = [T.sub.Lp,ChW,R]. (8)

[FIGURE 5 OMITTED]

ChW plant modeling

In this study, an equipment-performance-oriented plant model is proposed to calculate the plant power under predefined conditions. This model is based on a wire-to-water (WTW) plant efficiency concept. The plant total power can be calculated from the following formula:

[P.sub.Plant] = ([[xi].sub.CT] + [[xi].sub.CWP] + [[xi].sub.CHLR] + [[xi].sub.PPMP]) x [Q.sub.Plant,ChW] + [P.sub.SPMP]. (9)

Figure 5 is a flowchart of the ChW plant simulation. All of the variables on the left are the inputs, while the output is the plant total power. The plant model determines the plant total power consumption in response to a set of external parameters and a set of plant parameters.

This forward plant model can be set up easily and used for plant energy simulation. Since it is based on basic physical definitions and conservation laws, it has an explicit physical meaning. Its application is not restricted by the equipment number and sequencing strategies. All calculations are explicit expressions, and no iterations are required. One prerequisite is that the pumps are well sequenced and controlled such that the pump head and efficiency are around the normal operation point.

Chiller modeling

A Gordon-Ng model for vapor compression chillers with variable condenser flow is selected in this study. It can apply to unitary and large chillers operating under steady-state variable condenser flow conditions. This model is strictly applicable to inlet guide vane capacity control (as against cylinder unloading for reciprocating chillers or variable-frequency drive [VFD] for centrifugal chillers). The chiller motor power ([P.sub.CHLR,per]) can be induced from the linear form of the fundamental Gordon-Ng model for vapor compression chillers with variable condenser flow (Jiang and Reddy 2003). The chiller WTW efficiency (kW per ton) is

[xi]CHLR = [P.sub.CHLR,per] / [Q.sub.CHLR,per]. (10)

It is noted that the actual chiller ChW flow is also limited by the upper and lower limits of evaporator ChW flow rate. The upper limit is intended to prevent erosion, and the lower limit is to prevent freezing in the tubes.

Pump modeling

The general calculation formula of the pump power is

[P.sub.pump] = 0.746V x H x SG / 3,960[[eta].sub.all] (11)

The pump WTW efficiency is

[[xi].sub.pump] = [P.sub.pump] / [Q.sub.ChW] = 0.746[N.sub.pump] x V x H / 3,960[[eta].sub.all] x 24/[V.sub.ChW][DELTA][T.sub.ChW] (12)

The WTW efficiencies for PPMPs and condenser water pumps (CWPs) are

[[xi].sub.PPMP] = 0.004521 [H.sub.PPMP] / [[eta].sub.PPMP][DELTA][T.sub.ChW], (13)

[[xi].sub.CWP] = 0.0001884 [H.sub.CWP] / [[eta].sub.CWP] [V.sub.CW,per] / [Q.sub.ChW,per]. (14)

For SPMPs, the pump power is

[P.sub.SPMP] = 0.746 [V.sub.Lp_ChW] x 2.31 x ([DP.sub.Lp] + e[V.sup.2.sub.Lp_ChW]) / 3,960[[eta].sub.SPMP] (15)

Obviously, the energy consumption of SPMPs is subject to the loop-side operation and is not determined by plant operations.

CT modeling

The mass and heat transfer process in a CT is fairly complicated. The effectiveness model is the most popular model in CT simulations, but iterations are required to obtain a converged solution. To overcome this obstacle, a simple CT fan power regression model is proposed to calculate the tower WTW performance:

[P.sub.CT] / [Q.sub.CW] = [d.sub.1] + [d.sub.2] / [DELTA][T.sub.App] = [P.sub.CT] / (1+0.2843[[xi].sub.CHLR])[Q.sub.ChW], (16)

[[xi].sub.CT] = [P.sub.CT] / [Q.sub.ChW] = ([d.sub.1] + [d.sub.2] / [DELTA][T.sub.app])(1 + 0.2843[[xi].sub.CHLR]) (17)

[DELTA][T.sub.App] is the actual CT approach temperature, which is obtained from the following formula:

[DELTA][T.sub.App] = [T.sub.CT,CW,R] - [T.sub.wb]. (18)

Loop delta-T modeling

The loop-side performance plays an important role in the ChW-storage tank operation. It is subject to many factors, such as chiller ChW leaving temperature, cooling coil air leaving temperature, type of flow control valves, coil design parameters and degrading due to fouling, tertiary connection types, coil cooling load, air economizers, etc. Considering the difficulties in developing a physical model to simulate the loop delta-T, a linear model regressed from the trended data is used in this study:

[DELTA][T.sub.Lp] = [n.summation over (i=1)][h.sub.i][x.sub.i] + [h.sub.0], (19)

[V.sub.Lp,ChW] = 24[Q.sub.Lp,Chw,] / [DELTA][T.sub.Lp], (20)

where [x.sub.i] denotes the variables that could be the dominant factors of the loop delta-T model, such as ChW supply temperature, loop total cooling load, ambient DB and WB temperature, hour of the day, weekday or weekends, and month. The air system side parameters, such as coil air leaving temperature, total air flow rate, coil design delta-T, and sensible load ratio, are not included due to the diversity or unpredictability.

The exact form of the regression model may vary for different projects. It could be necessary to build different models to accommodate air-conditioning system operation changes at different seasons. A constant delta-T can be used in a rough, first-order simulation.

In this study, the loop ChW load [Q.sub.Lp,ChW] is assumed to be perfectly known. There are various kinds of methods proposed to estimate the cooling load of a building. For example, Sun et al. (2010) used a weather prediction model and a simplified building model to compute building cooling load, which works well for a single building. However, for a large ChW system serving tens or hundreds of various facilities, such as a university campus or an airport ChW system, it is not easy to use this model to estimate the total system cooling load. In this circumstance, the cooling load profile of the system in previous years could be a good source as the cooling load input. It is also possible to build a cooling load regression model based on the day type, weather, facility operation schedule, etc. and to use this model to predict the cooling load in the next year. Overestimating the load could result in a conservative strategy and a lower saving potential, while underestimating could force the plant to stage-on more chillers during the peak demand period and achieve less cost savings. To minimize the negative impacts of cooling load uncertainty on the operating strategy projection, it is possible to define several scenarios, such as high cooling load, average cooling load, and low cooling load. Based on the simulation results, operators can balance the risks and benefits and make decisions.

TES control strategy

According to the definition of a control strategy, it is essentially a tag given to a sequence of operating modes that covers a single cycle of the cool storage system. This cycle is one day in this study.

For a TOU rate, the full-storage control strategy can be stated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

The chiller-priority control can be stated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The storage-priority control strategy can be stated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

For different projects, new control strategies can be designed and implemented into this sub-model to realize specific operating intent and achieve even larger savings.

Summary and discussion

A ChW-storage system provides great opportunities to reduce the electrical energy consumption and operating costs. This article proposes a generic methodology for determining the optimal operating strategies for a ChW-storage system under a TOU electricity rate structure. It combines plant, TES tank, and loop together to achieve the optimization of the whole system.

This method is based on an investigation of multiple search paths performed month by month. The system operating strategies are classified based on the type of control strategies and the maximal number of chillers running during the off-peak and on-peak periods, which is like demand limiting. The tank inventory and state change are described with ChW volume and tank ChW charging or discharging flow rate, which avoid tank inventory inconsistency. Based on a WTW efficiency concept, a forward plant model is built to simulate the ChW plant power at each time step, and no iterations are involved. A regression model is selected to simulate the loop ChW supply and return temperature difference, which is critical for a ChW-storage system. The operating strategy with the lowest billing cost is selected as the optimal strategy for the current month. A plant optimization procedure with a GRG nonlinear solver is followed for the selected optimal operating strategy to further improve the whole system performance. The final results will be a table showing the control strategies, maximum number of chillers staged-on during the off-peak and on-peak periods, and optimal controlled variable set-points for plant operation for each month, an approach that is easy for the operators to follow.

This method considers the complexity of cooling load profiles, rate structures, weather conditions, and system performance, and it is easy to understand and follow. The sub-model-based structure makes it easy for users to modify, simplify, or replace individual sub-models without affecting others. The tank minimum level set-point can be adjusted to accommodate a conservative or an aggressive operating attitude. A forward plant model eliminates the iterations and ensures the simulation can be finished in a reasonable time. The application of this method will be illustrated with a practical project in Part II.

DOI: 10.1080/10789669.2011.564258

Nomenclature

AHU = air-handling unit

c = chiller model coefficients or heat capacity, kJ/kg x K

C = cost, $

CHLR = chiller

ChW = chilled water

ChWLT = chilled water leaving temperature, [degrees]F ([degrees]C)

CT = cooling tower

CW = condenser water

CWLT = condenser water leaving temperature, [degrees]F ([degrees]C)

CWP = condenser water pump

d = cooling tower model coefficients

DB = dry bulb

DP = differential pressure

e = loop hydraulic performance coefficient

FOM = figure-of-merit

GPM = gallons per minute

GRG = generalized reduced gradient

h = loop delta-T model coefficients

H = water head, ft

N = number

NLP = nonlinear programming

P = power, kW

PID = proportional-integral-differential

PLR = part-load ratio

PPMP = primary pump

PSV = pressure sustaining valve

Q = cooling load, ton

R = electricity energy or demand rate, $/kWh or $/kW

RTP = real-time-pricing

SG = specific gravity of the fluid

SPMP = secondary pump

t = hour

T = temperature, [degrees]F([degrees]C)

TES = thermal energy storage

TOU = time-of-use

U = volume, gallon

V = flow rate, GPM

VFD = variable-frequency drive

WB = wet bulb

WTW = wire-to-water

x = tank ChW level ratio or independent variables

[DELTA]t = time step, h

[DELTA]T = temperature difference, [degrees]F ([degrees]C)

Greek symbols

[eta] = efficiency

[xi] = wire-to-water efficiency, kW/ton

[phi] = figure-of-merit

Subscripts

App = approach

Cap = capacity

d = demand

db = dry bulb

e = energy

i = month

k = current hour

Lp = loop

max = maximum

min = minimum

mtr = motor

Opt = optimal

R = return

ref = reference

s = summer

S = supply

sp = set-point

sys = system

v = rate period

w = winter

wb = wet bulb

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Zhiqin Zhang, (1,2), * William D. Turner, (1) Qiang Chen, (2) Chen Xu, (3) and Song Deng (2)

(1) Department of Mechanical Engineering, Texas A&M University, College Station, TX 77943, USA

(2) Energy Systems Laboratory, Texas A&M University, 3581 TAMU, 214 Wisenbaker Engineering Research Center, Bizzel Street, College Station, Texas 77943-3581, USA

(3) VisionBEE, Austin, TX, USA

* Corresponding author e-mail: zhangzhiqin2010@gmail.com

Received September 20, 2010; accepted January 28, 2011

Zhiqin Zhang, is PhD Student and Research Assistant. William D. Turner, PhD, PE, is Professor. Qiang Chen, PE, Associate Member ASHRAE, is Research Engineer. Chen Xu, PE, Associate Member ASHRAE, is Project Manager. Song Deng, PE, Member ASHRAE, is Associate Director.
Table 1. Typical TOU rate structure.

 Winter billing Summer billing
Rate months months

Energy rate
On peak Re_w_on Re_s_on
Off peak Re_w_off Re_s_off
Demand rate
On peak Rd_w_on Rd_s_on
Off peak Rd_w_off Rd_s_off
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Author:Zhang, Zhiqin; Turner, William D.; Chen, Qiang; Xu, Chen; Deng, Song
Publication:HVAC & R Research
Date:Sep 1, 2011
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