Development of a Predictive Optimal Controller for Thermal Energy Storage Systems Gregor P. Henze Robert H. Dodiery Moncef Krartiz Joint Center for Energy Management CEAE Department, Campus Box 428 University of Colorado at Boulder Boulder, Colorado 80309{0428 January 15, 1997 Abstract
This paper describes the development and simulation of a predictive optimal controller for thermal energy storage systems. The `optimal' strategy minimizes the cost of operating the cooling plant over the simulation horizon. The particular case of a popular ice storage system (ice-on-coil with internal melt) has been investigated in a simulation environment. Various predictor models have been analyzed with respect to their performance in forecasting cooling load data and information on ambient conditions (dry-bulb and wet-bulb temperatures). The predictor model provides load and weather information to the optimal controller in discrete time steps. An optimal storage charging and discharging strategy is planned at every time step over a xed look-ahead time window utilizing newly available information. The rst action of the optimal sequence of actions is executed over the next time step and the planning process is repeated at every following time step. The e ect of the length of the planning horizon was investigated and it was found that short horizons on the order of one day are only marginally suboptimal relative to a strategy that is optimal over the entire simulation horizon of one week. Various utility rate structures were analyzed to cover a range of potential real-time pricing scenarios. The predictive optimal controller was then compared to conventional control heuristics. In summary, it was found that in the presence of complex rate structures the proposed controller has a signi cant performance bene t over the conventional controls while requiring only a simple predictor.
1 Introduction Thermal energy storage provides the opportunity to reduce the operating cost of the cooling plant in commercial buildings. Though ice storage technology as a popular implementation of thermal energy storage is the focus of this work, the methodology can easily be extended to other types of thermal energy storage.
1.1 Previous Work
Experience with operating ice storage systems shows that a signi cant shortfall of the current technology is the poor performance of the control strategies (Akbari and Sezgen (1992), Guven and Flynn (1992), Tran et al. (1989)). The best-known control mechanism for these systems is chiller-priority. This simple heuristic consists of using direct cooling (no storage usage) anytime during o -peak hours, and during on-peak hours E-Mail:
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(a) as long as the current cooling load can be met by a chiller that is downsized in capacity, or (b) while an imposed electrical demand limit is not yet exceeded by the cooling plant; otherwise the storage provides the remaining cooling capacity. In this study, the cooling plant consists of one chiller for both chilled-water and icemaking modes, pumps, fans, cooling tower, and all other equipment needed to provide cooling as sketched in Figure 3. Though chiller-priority can be an appropriate strategy under peak cooling load conditions, it generally makes poor use of the available storage during swing seasons. With chiller-priority being the most widely used control, the potential of ice storage systems is only partially harnessed. Recognizing the need for research in the area of optimal control for thermal storage systems, various approaches have been proposed. However, the term \optimal control" has been used rather loosely when referring to the control of ice storage systems. Spethmann (1989) states the optimization problem as a list of desirable features that includes the proper consideration of utility rates, cost of ice-making as a function of evaporator temperature, shape of cooling load pro le, and many more. Rather than nding the solution to a strictly variational problem, optimal control is treated as the problem of developing a knowledge base for a real-time expert system. Similarly, Knebel (1990) interprets optimal control for an ice harvesting system as the correct variation of build and defrost time as a function of compressor capacity and saturated discharge pressure of the refrigerant. Braun (1992) and Drees (1994), on the other hand, carried out simulations that are in accordance with the strict mathematical de nition of optimal control. Assuming known load pro les, Braun carried out daily simulations comparing chiller-priority, storage-priority, and optimal control, where the latter was based on the minimization of either energy charges or peak demand charges. Drees, in addition to minimizing either daily energy or demand charges, investigated an objective function that minimizes energy cost while preventing the demand cost from exceeding a prede ned target. He also developed a heuristic control strategy that is simple to implement yet approaches optimal performance. Kintner-Meyer and Emery (1995) investigated the bene ts of an optimal control that minimizes daily (i.e., 24 h) operating cost for a large oce building by adjusting indoor air temperature and humidity setpoints as well as cooling plant operating parameters. Two sources of thermal energy storage were considered: an ice storage tank (ice-on-coil with internal melt) and the thermal capacitance of the building mass (heavy, light, and ultra-light construction). The investigation therefore links the problem of dynamic building control (building pre-cooling or pre-heating) with the problem of cool storage control. Thermal building response and loads are part of the problem of HVAC control with TES rather than external constraints on the optimization. The study therefore e ectively compares the individual and combined e ects of building internal mass and cool storage on load shifting and peak power reduction. They found that signi cant operating cost savings and electrical demand reduction can be achieved, compared to the basecase of using direct cooling only, through early-morning ventilation utilizing free cooling and by shifting cooling loads to o -peak periods. The investigators concluded that optimal building control is most e ective in dry climates with large diurnal temperature swings, in the presence of utility rates strongly encouraging load-shifting, and when cool storage systems allow more e ective load-shifting than building pre-cooling alone. Driven by the need for a better understanding of the behavior of ice storage systems in general, a simulation environment had been developed by Krarti et al. (1995) that allows the investigation of a wide range of key parameters in uencing the system's operating cost. Within this environment, the optimal control strategy that minimizes the total electricity cost combining energy and demand charges was developed and validated. Based on building cooling loads and weather data, the optimal control strategy properly accounted for the e ects of all environmental variables including utility rate structure and cooling plant characteristics. This optimal strategy was used as a reference against which conventional control strategies were compared. Three conventional control strategies were modeled: Chillerpriority, constant-proportion, and storage-priority control. Chiller-priority and constant-proportion control are both instantaneous. Like optimal control, storage-priority control requires short-term load forecasting. As a fundamental assumption, perfect predictions of weather and load variables were assumed for both storagepriority and optimal control in the study of Krarti et al. (1995). In particular, Krarti et al. (1995) found that under favorable conditions such as high cost incentives to shift loads to o -peak hours or a small energy penalty for operating the chiller in icemaking mode, an ice 2
storage system with the conventional controls can provide considerable cost savings. However, as soon as the utility incentive for load shifting is reduced or the use of the ice storage is associated with a signi cant increase in total energy use, only optimal control will be able to yield energy cost savings. In all cases, optimal control was shown to succeed in reducing operating cost, even under the most adverse conditions. Among the conventional control strategies, storage-priority performs the best under the premise that the cooling loads over the next on-peak period are always perfectly known. Under real-world conditions, however, the future loads even a few hours into the future cannot be perfectly predicted.
1.2 Scope of Present Work
The present work investigates the potential bene ts of optimal control for ice storage systems under real-time pricing. In a deregulated environment, the cost of energy will most likely change on an hourly basis. The work rests on the assumption that real-time pricing utility rates, unlike current rate structures, contain no demand charges, the cost of electrical energy varies smoothly in daily cycles (though discretely from hour to hour) and will only be known over a short time window, e.g. the next 24 hours. As the results by Krarti et al. (1995) indicated, optimal control is signi cantly superior when load-shifting incentives are weak. When the cost of energy changes in a smooth fashion rather than the current step changes from low o -peak to high on-peak rates, optimal control is expected to better weigh these varying cost e ects since the conventional controls all rely on an unambiguous separation between on-peak and o -peak periods.
2 System Modeling
2.1 Building Types
The building investigated in this study is a large 20- oor oce building. The cooling load pro le and pro le of ambient climatic conditions of this site is part of a unique commercial cooling load library established by the Energy Center of Wisconsin (ECW, 1994). The site is located in the state of Wisconsin. The building has a oor area of 350,000 ft2 , 2,800 employees, 85 hours of operation, and is located near a lake. The beginning and end of the occupied periods in the particular sites were not available from the source. Yet, for the controllers formulated in this investigation, this is not an essential information and the times of the on-peak and o -peak periods suce. The weekly cooling load pro le shown in Figure 1 is that during which the peak cooling load occurs. This peak cooling load occurred in the month of August 1990. Similarly, Figure 2 shows a graph of the weekly wet-bulb temperature pro le for the investigated site for the same time period as the cooling loads. The realistic cooling plant model used in the presented work incorporates a water-cooled condenser whose behavior is better modeled using wet-bulb rather than dry-bulb temperatures.
2.2 Basecase
To make meaningful comparisons, the basecase refers to that cooling system that experiences the same cooling load and weather pro les and uses the same chiller and air-handler subject to the same utility rate structure as the corresponding ice storage system. The only di erences are that (i) the chiller is sized to fully meet the peak cooling load, i.e. it is not downsized as for the case of chiller-priority, and (ii) there is no ice storage system available. It represents the standard case of a cooling system without cool storage.
2.3 Thermal Storage System
The most important feature of any storage system is its ability to bridge a temporal gap between supply and demand. In a cooling plant without thermal storage, the cooling load is met through direct cooling; the corresponding power consumption of the plant cannot be postponed to later periods when electricity is cheaper. In a thermal energy storage system, the temporal occurrence of electrical cooling-related loads can 3
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0
0
24
48 72 96 120 Time [hrs] starting at Monday 00:00
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Figure 1: Cooling load pro le of investigated large 20- oor oce building.
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Wet-Bulb Temperature [deg F]
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48 72 96 120 Time [hrs] beginning Monday 00:00
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Figure 2: Wet-bulb temperature pro le of investigated large 20- oor oce building.
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be separated from that of the thermal (cooling) loads. Consequently, there is a exibility in the choice of the cooling source: either direct cooling from the chiller, or discharging the storage tank, or a combination of those. At night, when cooling is often not required, the tank is recharged. To describe the state of the storage, only one state variable, the state-of-charge x of the storage tank, needs be introduced. The scalar control variable u is de ned as the rate of change of the state-of-charge x, u = dx dt . At any time, the question is to either charge, discharge or remain inactive as expressed by the control variable u. State transitions of the thermal storage system regardless of the selected ice storage model are described in discrete time by t (1) xk+1 = xk + uk SCAP subject to the state constraints
xmin xk xmax
(2)
umin;k uk umax;k :
(3)
where SCAP is the capacity of the storage; the control variable u is subject to the constraints
where for all ice storage models, umin;k is the negative value of the maximal discharging rate at time k, and umax;k is the maximal charging rate at time k. Several components a ect these control constraints. As an example, the bounds on the action variable are a function of the state-of-charge x of the tank, i.e. controls u that lead to states-of-charge less than xmin or greater than xmax are inadmissible. The nal control constraints, however, depend on the underlying ice storage model. In case of the basic plant model used in this study, the charge and discharge capacities depend only on available ice inventory and current cooling load while for the realistic plant model charge and discharge capacity are modeled as a function of the ice storage type characteristics and ice inventory. Both the basic and realistic models will be described below.
2.4 Design Parameters
Certain design choices had to be made to x the cooling plant capacities. The ice storage tank is designed to store up to four times the peak cooling load Qpeak over the extent of one hour (t), i.e.
SCAP = 4Qpeak t;
(4)
while the chiller is sized to fully meet the peak load, i.e.
CCAP = Qpeak :
(5)
For the control strategy chiller-priority, however, the chiller is downsized by 25%, i.e. CCAP = 0:75Qpeak .
2.5 Cooling Plant
Two cooling plant models with ice storage systems of disparate complexity are used throughout the present work. Both models are steady-state and the dynamics associated with changing from one charging/discharging rate at the end of each hour to another cannot be captured. However, the time increment of one hour will be signi cantly larger than the time constant of the cooling plant including the ice storage system. 2.5.1
Basic Plant Model
In this simplistic model, the cooling plant is assumed to operate with a constant eciency of EIR(chw) in units of [kW/ton] in the chilled-water mode and with a reduced eciency EIR(ice) in the ice-making mode, i.e. EIR(chw) < EIR(ice) . The higher EIR in icemaking mode re ects the increased power consumption when 5
providing cooling at subfreezing temperatures. These values lump together the eciencies of all components of the cooling plant. The total power consumption Pk of the basic plant model is ch;k EIR(ice) if u > 0, i.e. charging Pk = Q (6) Qch;k EIR(chw) if u 0, i.e. discharging where the chiller load Qch;k is de ned as the sum of the cooling load Qk and the discharging/charging rate uk , Qch;k = Qk + uk : (7) For the basic plant model, the charge and discharge capacities depend only on available ice inventory and current cooling load. The constraints on the control variable for the basic plant model are formulated as (8) umin = maxf?Qk ; uxk+1=xmin g and (9) umax = minfCCAP ? Qk ; uxk+1 =xmax g 0 where uxk+1 =x denotes the control action u that leads to a state-of-charge x when operating the cooling system over the next hour (k ! k + 1) at exactly that control u. Thus, no actions can be taken that would lead to states-of-charge outside the limits. Furthermore, the ( xed) capacity of the chiller CCAP minus the current load Qk de nes the maximal charging rate umax;k , i.e. how much can be charged after meeting the load. 0
2.5.2
Realistic Plant Model
The realistic cooling plant model used in this study is a modular steady-state model that computes the power consumption of each plant component (Krarti et al. (1995)). The correlations used in the plant model are based on the building simulation program BLAST (1994). The components include an air-handling unit, an ice storage system, a chiller, a cooling tower, and all required pumps and fans as illustrated in Figure 3. The modularity allows one to easily switch between one of three ice storage system types (ice-on-coil with internal melt, ice-on-coil with external melt, and a dynamic ice harvester) as well as between one of three chiller compressor types (centrifugal, reciprocating, and screw compressors). The plant model determines the component power consumptions in response to a set of external parameters and a set of plant parameters. The external parameters are the cooling load Qk at hour k, the ambient wetbulb temperature at that hour TWB;k , the state-of-charge xk , and the charging/discharging rate uk . In addition, a set of plant parameters governs the operation of the cooling plant. For the realistic plant model used in this analysis, the plant parameters include either one or two temperature setpoints of the cooling plant depending on the mode of operation. These cooling plant temperature setpoints are the chilled water supply temperature leaving the evaporator, the chilled water supply temperature entering the air handling unit, and the supply air temperature and they are adjusted in a separate optimization step (plant optimization) so that the total power consumption for the current set of external parameters is minimized. Only the simultaneous cost and plant optimization allows the determination of an optimal control trajectory in which each charging/discharging control is associated with the minimal attainable cooling plant power consumption. When evaluating conventional control strategies, the plant power consumptions will also be the smallest attainable for the chosen control action u under the given conditions. The plant optimization is described in detail by Krarti et al. (1995). In case of a realistic cooling plant model, constraints on the heat transfer capabilities of the thermal storage system further restrict the admissible values of u. These additional constraints include the ability to absorb or reject heat as expressed by umin;TES (x) and umax;TES (x), respectively. These limits depend on the selected thermal storage system type and the state-of-charge x. The constraints on the control variable for the realistic plant model are formulated as (10) umin = maxf?Qk ; uxk+1=xmin ; umin;TES (x)g 6
and
umax = minfuxk+1=xmax ; umax;TES (x)g
(11) Initially, the realistic plant model determines the peak ice storage charging rate umax;TES (x) and peak discharging rate umin;TES (x) as the upper and lower limit on the control variable u. The exact calculation of umax;TES (x) and umin;TES (x) requires the solution of multiple nonlinear equations governing the heat transfer between the ice storage, chiller, and air handler. The maximal possible discharge rate umin;TES (x) occurs when the cooling load is met the storage device alone at the maximal rate when both loads, the ice storage load Q_ ice and air handler load Q_ l , are still balanced, i.e. Q_ ice = Q_ l . The maximal charge umax;TES (x) can be found when the chiller is making ice and meeting the cooling load at the maximal possible rate, i.e. as long as Q_ ch = Q_ l + Q_ ice is satis ed. The chiller performance is modeled according to BLAST (1994) using quadratic expressions with coecients taken from manufacturers' data. For details on the determination of the charge and discharge capacity limits for ice-on-coil systems with external and internal melt as well as ice harvesters refer to Krarti et al. (1995). To cooling tower
Condenser VCRC Cycle
Expansion Valve
Compressor
Evaporator
Ice Storage System
Tfr
Tfr
Air Handling Unit
To building cooling load
Figure 3: Diagram of cooling plant con guration.
3 Conventional Control Strategies This study analyzes various control strategies for ice storage systems which are a popular choice of thermal energy storage technology. However, the controls are not necessarily limited to the application of ice storage systems. Two instantaneous and one predictive conventional control strategies have been investigated. The instantaneous strategies only require the knowledge of current quantities such as cooling load and control limits.
3.1 Chiller-Priority Control
The simplest of the existing control strategies for thermal energy storage is chiller-priority control. Here, the chiller runs continuously under conventional chiller control (direct cooling) possibly subject to a demand-limit 7
while the storage provides the remaining cooling capacity if required.
8 < umax;k if k is o -peak if k is on-peak and CCAPk Qk (12) uk = : 0 umax;k if k is on-peak and CCAPk < Qk (Note: umax < 0 in this case) where uk is the charging/discharging rate, CCAPk is the reduced (75%) chiller capacity, and Qk is the cooling load each taken at hour k. The simplicity lies in the fact that the conventional chiller control is not altered yielding high chiller eciencies and a smooth demand curve, however, with the disadvantage that the meltdown of the ice is not controlled to allow for maximal demand reduction. In addition, the time-of-day dependent energy rate structure in [$/kWh] is not accounted for, i.e. load-shifting performance is inadequate.
3.2 Constant-Proportion Control
This control strategy implies that the storage meets a constant fraction fQ of the cooling load under all conditions. Thus, neither the chiller nor the storage have priority in providing cooling. This simple control strategy provides a greater demand reduction than chiller-priority control since the chiller capacity fraction used for a particular month will track the fraction of the annual cooling design load the building experiences during that month. For example, in a month in which 50% of the annual design load occurs, only 50% of the chiller capacity will be requested.
uk =
umax;k
if k is o -peak maxf?fQQk ; umin;k g if k is on-peak
(13)
Constant-proportion control is rather easy to implement by assigning a xed fraction of the total temperature di erence between brine supply and return ow to be realized by the storage and the remainder by the chiller. Finding the best load fraction fQ for each application is a matter of trial and error. Caution should be exercised that the chiller can always meet the remaining load fraction, i.e. (1 ? fQ )Qmax CCAPnom or in other terms
nom ; fQ 1 ? CCAP Q peak
where CCAPnom is the nominal chiller capacity.
3.3 Storage-Priority Control
As the name re ects, storage-priority control requires melting as much ice as possible during the on-peak period. It is generally de ned as that control strategy that aims at fully discharging the available storage capacity over the next on-peak period. Thus, the simultaneous operation of the chiller and the storage and the terminal state-of-charge is speci ed, yet not how this is accomplished in detail; see Tamblyn (1985). In the present work the following implementation is suggested. The chiller is base loaded during o -peak hours to recharge the ice storage for the next on-peak period. The chiller operates in one of two modes during the on-peak period. The chiller operates either in Mode 0 at a reduced capacity A0 in parallel with the storage during on-peak hours so that at the end of the on-peak period the storage inventory is just depleted. If this is not possible without prematurely depleting storage, the control switches to Mode 1 and the storage provides a constant proportion A1 of the load in each on-peak hour similar to constant-proportion control. Both parameters, A0 and A1 are determined from predicted cooling loads over the next on-peak period. The on-peak period used throughout the analysis regardless of whether time-of-use or real-time pricing rates are used, is from 10 a.m. until 6 p.m. during weekdays. All other hours are o -peak. 8
The tank discharge/charge rates are determined by
8 < uk = :
umax;k
if k is o -peak
A0 ? Q k if k is on-peak and discharge mode is 0 maxf?A1 Qk ; umin;k g if k is on-peak and discharge mode is 1
(14)
The chiller capacity A0 is computed from
(N ?1 ) X SCAP 1 Qk +i ? xk t A0 = N (15) i=0 where N is the number of hours in the on-peak period and k0 is the rst hour therein. The constant proportion A1 in turn is determined from SCAP A1 = PxNk?1 t (16) i=0 Qk +i Refer to Krarti et al. (1995) for a more detailed description of storage-priority control and graphs of the performance and behavior of chiller-priority, constant-proportion, and storage-priority controls. 0
0
0
0
4 Optimal Control Optimal control in the current context is that sequence of controls that minimizes the operating cost of the cooling plant over the simulation period
J = J (u1 ; u2 ; : : : ; uK ) =
K X k=0
rk Pk t
(17)
where K is the total number of hours in the simulation period, rk is the cost of electrical energy at hour k, and Pk is the cooling plant electrical power consumption. The sequence frk gKk=0 is the cost of electrical energy as driven by real-time pricing in a deregulated utility environment. Several periodic utility rate structures with varying amplitude were investigated as shown in Figure 4. Note that rk is only an energy charge; in the real-time pricing scenario we have investigated, there are no demand charges. For all rate structures and for all control strategies, on-peak period is de ned as the time between 10 a.m. and 6 p.m. Monday through Friday with o -peak period being the remaining hours and weekend. All three conventional controls require information on the beginning and end of on-peak periods to function properly as indicated by Equations 12, 13, and 14. Times of actual occupancy are not required, since the discharge rates are adjusted as a function of the current cooling load. The real-time pricing scenarios in this work are chosen for illustrative purposes and no claim is made that these rate structures will be typical once utilities are deregulated. Rather, the examined rate structures represent idealized pro les of what the authors consider probable realizations.
4.1 Dynamic Programming
The task of minimizing operating cost J is framed as a sequential decision-making process of decision variable u: The optimization technique called dynamic programming, commonly used for this type of problems, was rst formally introduced by the mathematician Richard Bellman in 1957. Bellman's Principle of Optimality states that the optimal solution to a K -step process must come from the optimal solution of an K ? 1-step process that is based on the optimal outcome of the rst step. The solution of one K -step process will thus be found recursively by optimizing K single-step processes. 9
0.18 Rate Structure 1 Rate Structure 2 Rate Structure 3
0.16
Cost of Electricity [$/kWh]
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
0
6
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Figure 4: Rate Structures used in the study. The basic optimization model has two fundamental features: (i) an underlying discrete-time dynamic system, and (ii) an incrementally additive cost function. The dynamic system has the general form xk+1 = f (xk ; uk ; wk ) (18) where x is the state of the system, u is the control, and w is a random disturbance. Consider a process in which the transition from one state xj at time j through the control uj is associated with the instantaneous cost l(xj ; uj ; wj ): (19) Then the total expected cost of starting at time k and ending at time K can be expressed by
J=
K X j =k
E [l(xj ; uj ; wj )]
(20)
where the expectation in this and other equations is taken with respect to the disturbance wj . It is assumed there exists a sequence of controls u that minimizes J , thus there exists for each state x and time k an optimal cost-to-go function,
8 9 K
