1. Introduction
Vapour-compression refrigeration systems represent the most used technique worldwide to provide cooling power, including freezing, cooling, and air conditioning applications. The amount of energy involved in such processes is huge, both for industrial and commercial/residential purposes [
1]. It is reported that up to 30% of total energy needs around the world are due to Heating, Ventilating, and Air Conditioning (HVAC) systems, among which refrigeration processes represent a significant share [
2].
Energy efficiency of current mechanical compression refrigeration systems has been highly improved over time, while their environmental impact has been reduced due to diverse factors. Firstly, more effective design of heat exchangers, variable-speed compressors, and electronic expansion valves (EEV) have improved heat transfer efficiency and controllability. Secondly, more efficient control strategies have been applied to satisfy cooling demand while rejecting disturbances and achieving the highest energy efficiency possible with a direct environmental impact. For instance, Jain and Alleyne developed a model predictive controller using a dynamic exergy-based objective function to maximize exergetic efficiency of a basic refrigeration cycle [
3]. Concerning HVAC systems in buildings, Espejel-Blanco et al. have recently presented an HVAC temperature control system based on the Predicted Mean Vote (PMV) index calculation, which combines the values of humidity and temperature to define comfort zones and achieves energy savings ranging from 33% to 44% against the built-in control of the HVAC equipment [
4]. Kou et al. develop both model-based and data-driven HVAC control strategies to determine the optimal control actions for HVAC systems, highlighting a significant trade-off between electricity costs and computational speed between both strategies [
5]. Furthermore, Macieira et al. have also presented an energy management model for HVAC control based on reinforcement learning, to manage the HVAC units in smart buildings according to the prediction of users, current environmental context, and current energy prices [
6].
In recent years, a novel line of research regarding cold-energy management has been introduced which involves the addition of thermal energy storage (TES) systems to the standard refrigeration cycle. Accordingly, cooling production and demand can be decoupled, since excess cold energy may be stored in the TES system, being released when required, as common storage strategies applied to solar plants [
7]. This setup makes it possible to use lower capacity systems, since they need not be oversized to face peak-demand periods [
8], and thus the investment cost may be reduced. Moreover, the cycle may be operated in more favourable terms, which improves efficiency and reduces energy usage. Furthermore, it is possible to schedule the cooling production according to the variable cost of electricity given by the energy market. Thus, an optimal strategy (peak-shifting) would be to produce as much cold energy as possible during the low-priced periods, while storing the excess in the TES system, and to produce as little as possible during the high-priced periods, releasing the cold energy previously stored [
9].
Concerning the design of the storage system, most commercial and developing systems rely on phase change materials (PCM) [
10]. The main reason is that their thermodynamic properties are more suitable for energy storage than those of sensible-heat materials [
11]. PCM are able to store larger amount of energy per unit volume and their temperature remains approximately constant in latent state, which improves heat transfer efficiency. Concerning domestic refrigerators, Bista et al. published a thorough review [
12], regarding PCM selection and placement inside the refrigerator, where the use of PCM is shown to increase the Coefficient of Performance (COP) with a relatively significant margin. Among the PCM-based systems, there are several factors defining the storage configuration, such as the PCM encapsulation and the heat exchanger layout where the PCM and the heat transfer fluid (HTF) meet, the packed bed technology being one of the most used configurations [
13]. Regarding the latter, Berdja et al. proposed a novel approach to optimize the dimensions of the packed bed heat exchanger, according to the technical features and operating conditions of a standard refrigerator [
14].
Many works in the literature have addressed cold-energy management of refrigeration systems with energy storage. For instance, three works by Wang et al. cover the design [
15], modelling [
16], and control [
17] of a large-scale HVAC plant, where a ring of PCM-based TES tanks acts as a cold-energy reservoir. Control policies are proposed to activate/deactivate the TES tanks, seeking high performance. Other works implement management strategies based on exergy analysis, where a combined use of the different PCM modules is applied [
18]. Furthermore, model predictive control (MPC) also represents a successful strategy. Indeed, it has been applied for HVAC without energy storage [
19,
20], but the potential of predictive strategies for TES-backed-up systems is much higher. For instance, Shafiei et al. [
21] propose a MPC framework to minimize the divergence in electric energy usage with respect to a given reference, in a large-scale refrigeration plant. For that purpose, energy collected by and retrieved from the TES is estimated, and an optimization problem is formulated by introducing the reference of the evaporation temperature as a virtual control variable. Shafiei et al. [
22] also apply predictive control to TES-backed-up refrigerated freight transport, where the TES tank is embedded in parallel with the refrigeration cycle. The prescribed delivery route profile, along with traffic and environmental information, are incorporated into the algorithm to predict the cooling demand. Furthermore, Schalbart et al. present an MPC strategy to the management of an ice-cream warehouse refrigeration system coupled to a TES tank [
23]. The controller is based on a steady-state refrigeration cycle model and energy balances on the TES tank and the warehouse.
In this work, a novel hybrid configuration, that comprises a PCM-based TES tank especially designed to complement an existing refrigeration system, is analysed from the point of view of energy management. The layout, shown in
Figure 1, is slightly different to that usually applied in the aforementioned works, due to the features of the original refrigeration facility.
As depicted in
Figure 1, the fluid to be cooled (hereinafter referred to as secondary fluid) is pumped from a tank, which represents the refrigerated chamber, both to the evaporator and to the TES tank, and it is then recirculated to the chamber. An electric resistance is used at this tank to produce heat and simulate the cooling demand, which must be satisfied by both secondary fluid streams. The vapour-compression cycle transfers cold energy to the evaporator by cooling the secondary fluid, but the refrigerant also circulates through the TES tank while charging it, being the latter the cold HTF in this case. The secondary fluid also circulates through the TES tank while discharging it, and thus the warm HTF does not match the cold HTF, as a very relevant difference with respect to the packed bed technology. The layout of the TES tank comprises a number of PCM cylinders and two bundles of pipes, one corresponding to the refrigerant and the other to the secondary fluid, all of them bathed in the so-called intermediate fluid. This TES setup is very similar to that described in a recent work by Bejarano et al. [
24], the PCM encapsulation being the only relevant difference. To the authors’ knowledge, this setup is novel and the cold-energy management and economic potential offered to industrial refrigeration facilities is first analysed in this work, which represents one of the main contributions of the article. This configuration involves several operating modes, according to the manipulation of the expansion valves and pumps, that enable/disable the different fluid streams and make the system hybrid, in addition to its inherent non-linear features.
As previously mentioned, cold-energy storage allows decoupling of cooling demand and production. Moreover, given the parallel arrangement between the evaporator and the TES tank, the cooling demand at the refrigerated chamber may be satisfied using both secondary fluid streams. It implies that the TES-backed-up refrigeration system is able to satisfy peak cooling demand that might be infeasible for the original standard refrigeration cycle, due to the double contribution provided by both secondary fluid streams. The economic and feasibility potential offered by the addition of the TES tank to the original refrigeration cycle is explored by proposing a scheduling and control strategy based on non-linear MPC to satisfy the cooling demand while minimizing the daily operating cost.
The main contribution of this work regarding the scheduling is the application of non-linear model predictive control (NMPC) to this hybrid system, where the computational cost of the prediction model is also considered. A non-linear model describing the dominant dynamics of the TES tank is proposed to be used as the prediction model. The latter is developed from the frequency analysis performed in the work by Rodríguez et al. [
25], where it is stated that two very different time scales arise in the combined system: one related to the fastest dynamics of the refrigeration cycle, and another, slower one, related to heat transfer within the TES tank. In that work, a detailed model of the TES-backed-up refrigeration cycle was presented, focusing on the faster dynamics caused by the refrigerant circulation, which must be integrated using a small sampling time, in the order of a few seconds. However, such complexity is not affordable or even necessary when addressing the scheduling stage, according to the desired sampling time of several minutes, given the slower dynamics related to heat transfer within the TES tank. Therefore, in this work a simplified non-linear model is proposed, focusing on the dominant dynamics related to the TES tank, which turns out to be far from trivial but more suitable to model-based predictive strategies, especially concerning computational load. This is one of the main contributions of the work, since the reduced computational cost of the prediction model allows direct application of a non-linear model predictive control strategy while ensuring reasonable computation times. The performance of the proposed scheduler for the TES-backed-up system is compared in simulation with the refrigeration cycle without energy storage, while an analysis on the operating cost and constraint meeting is also performed, even when considering significant parametric uncertainty in the prediction model.
Therefore, the main contributions of the work are summarised below:
A novel setup of a TES-backed-up refrigeration system is presented, where the TES tank is inserted within the refrigeration cycle, causing the heat transfer fluid to differ between charging and discharging processes. This is an essential and relevant difference with respect to the dominant packed bed technology [
13,
14].
A layered NMPC-based scheduling and control strategy is applied to the TES-backed-up refrigeration system, which explores the economic and feasibility potential of such a setup, where the control of the compressor and expansion valves is directly affected by the embedding of the TES tank within the refrigeration cycle, in contrast with the packed bed technology, where the refrigerant circulation is not affected by the state of the storage tank [
13]. Thus, the existing control strategies previously detailed [
17,
18,
21,
22,
23] cannot be applied to the configuration under study.
A simplified and computationally efficient non-linear prediction model describing the dominant dynamics of the system is presented, which allows the application of the previously mentioned non-linear predictive strategy with reasonable computational load, when compared with the accurate but computationally expensive existing dynamic model [
25].
The work is organised as follows.
Section 2 describes the main characteristics of the designed TES tank and its embedding in the existing facility.
Section 3 addresses the modelling of the combined system, focusing on the dominant dynamics related to heat transfer within the TES tank. The overall control strategy is presented in
Section 4, focusing on the NMPC-based scheduling algorithm.
Section 5 describes a case study for a demanding load profile, where the need for operating mode scheduling is justified according to the power constraints. The economic performance and demand satisfaction of the scheduling strategy are compared to that of the original cycle without energy storage in simulation. Finally,
Section 6 summarises the main conclusions and some future work is proposed.
5. Case Study
This section is devoted to the analysis of a case study, focusing on the scheduling problem. A challenging load profile is proposed, where the need for operating mode scheduling is highlighted. The presented simulation results compare the performance of the proposed scheduling strategy to a that of the same refrigeration cycle without TES, regarding economic cost and constraint meeting. A sensitivity analysis is also performed, where the performance of the proposed strategy is assessed when significant parametric uncertainty in the prediction model is considered.
5.1. Demand Profile
Consider the cooling demand profile
represented in
Figure 7. The shape could be similar to a typical supermarket load profile, but the specific power values have been tailored to the system under study described in
Section 2.
Please notice that the time window has been set to 12 h instead of considering a complete day due to the maximum charging/discharging periods for which the TES tank has been designed. As stated in [
24], the latter has been devised to ensure full charge/discharge in 3 or 4 h periods at full charging/discharging power, that were regarded as most desirable due to the research aim of the refrigeration facility. Therefore, longer charging/discharging periods cannot be fulfilled by this TES tank, which would be necessary to address actual 24-h time periods. However, notice that it is only a scaling factor, since, provided that the TES tank volume was high enough, 24-h time periods could be addressed following the same control strategy.
5.2. Cooling Power Ranges
Table 4 gathers the maximum and minimum values of the different cooling powers for the system under study [
25], regarding operating modes 1 to 4, which have been justified in
Section 3.2 to be the simplest and most likely to be scheduled. Notice that a minimum admissible value
= 2 °C has been imposed on the degree of superheating when obtaining the cooling power limits in mode 1.
As discussed in [
25], most cooling power limits depend on
, since the cylindrical shell in sensible zone grows as the TES is charged/discharged. This growth involves higher thermal resistance, that modifies the achievable cooling powers as
evolves. Power ranges when the freezing/melting boundary is located approximately at the cylinder edge, halfway, and at the centre are detailed in
Table 4.
It must be remarked that, in the case of mode 2, the cooling demand is provided exclusively at the evaporator, and thus the TES tank is not involved and that is why the cooling power range does not depend on the freezing/melting boundary location. However, in mode 4,
depends on the
charge ratio, which decreases as the TES is discharged. Operating mode 3 is merely a combination of the previously described modes, where two independent secondary fluid circuits are used, and thus the cooling power ranges are the same as those presented for modes 2 and 4. Nevertheless, when operating in mode 1,
and
are strongly correlated, because the refrigerant circulates through the evaporator and the TES tank, then both flows merge at the compressor intake (point A in
Figure 1). Therefore, the ranges indicated in
Table 4 are not completely rigorous, since they include some unachievable points. As an illustrative example,
Figure 8 shows the steady-state map between
and
when the freezing boundary locates at the PCM cylinder centre, where the continuous lines represent the overall limits and the small crosses indicate a number of steady-state achievable points.
Similar plots have been obtained for different locations of the freezing boundary, which are not included here for the sake of brevity. Notice that the cooling demand in operating mode 1 must be completely satisfied by , since is zero. Therefore, the maximum charging power is shown to be constrained by the cooling demand.
5.3. Energy Price
Realistic energy price evolution throughout the day is applied to the optimization problem.
Figure 9 depicts the electricity prices corresponding to a given day (29 May 2019) in the Spanish spot market [
31].
These prices, once scaled, are considered as the cost of the cooling power production at the evaporator, in Equation (22), as well as the cost of the TES charging power, . However, is assumed to be zero, given that no extra cost is involved when using the previously produced cold energy, beyond the electric power devoted to pumping the secondary fluid, which is not considered in the objective function detailed in Equation (22).
5.4. Operating Mode Scheduling
If the demand profile shown in
Figure 7 is analysed together with the cooling power constraints shown in
Table 4 and the energy prices shown in
Figure 9, it can be observed that, approximately, during the first 5 h, the cooling demand
can be satisfied by using only
, given the power ranges for modes 1 and 2 presented in
Table 4. It may be interesting to charge the TES tank during this period, taking advantage of the low energy price, but in any case the maximum charging power is constrained by the cooling demand, which must be satisfied in mode 1 by the refrigeration cycle. However, from
5 h, the cooling demand
is too high to be satisfied only by
, being necessary to turn to
to complement the cooling power produced at the evaporator. Then, during the peak period until
8 h, the TES tank is intended to be discharged, avoiding as much as possible the direct cooling power production due to the high price corresponding to this period. That is when the high
charge ratio achieved previously comes handy. Finally, at the end of the day, the cooling demand is again low enough to allow modes 1 and 2 to be scheduled, thus the TES tank might be also charged during this off-peak period.
In the light of the previous analysis, a certain operating mode scheduling must be proposed to make use of the TES tank as a cold-energy storage unit, as efficiently as possible, according to economic criteria. This operating mode scheduling affects the optimization problem posed in
Section 4.3 through the feasibility constraints described in (
19), as schematically shown in
Figure 10.
Figure 11 shows a proposed operating mode scheduling for the specific demand profile introduced in
Figure 7, according to the power constraints analysed in
Section 5.2 and the energy price profile shown in
Figure 9.
As observed in
Figure 11, the TES tank is scheduled to be charged during the off-peak periods at the beginning and at the end of the day, when the cost of cooling power production is lower. Given that the demand is low but not zero during these periods, operating mode 1 is scheduled, in such a way that the cooling demand is satisfied exclusively at the evaporator. Furthermore, the cold-energy stored in the TES tank is scheduled to be released during the peak period, complementing the cooling power provided to the secondary fluid at the evaporator to meet the cooling demand (operating mode 3). Two intermediate periods when the TES tank is not charged nor discharged, but in stand-by (operating mode 2), have been inserted between the charging and discharging periods, given that the cooling demand can be satisfied at the evaporator and it is certainly more interesting from an economic point of view to ensure the cooling demand satisfaction during all the peak period, when the contribution of the TES tank is imperative. Moreover, once the peak period is finished, it might be more interesting to concentrate the TES tank charging during the central hours of the off-peak period, when the economic cost of cooling power production is lowest.
5.5. Simulation Results
Some simulation results of the proposed scheduling strategy are presented in this subsection, where they are also compared to the performance of the original refrigeration system without energy storage, regarding both economic cost and constraint meeting.
Given the cooling demand shown in
Figure 7 and the energy price profile shown in
Figure 9, the references of the cooling powers are computed every hour. However, the dynamics related to heat transfer within the TES tank require an internal sampling time of the prediction model of 10 min. The latter could be reduced to increase accuracy, at the expense of greater computational load of the predictive scheduling strategy. Safety limits
= 0.05 and
= 0.95 have been considered, while a prediction horizon
= 4 is set. At the initial state of the TES tank, the intermediate fluid is assumed to be in thermal equilibrium with the PCM cylinders (
), whereas the initial enthalpy distribution of the latter is such that
= 0.35.
Figure 12 shows the performance of both the proposed scheduling strategy for the TES-backed-up system and the refrigeration cycle without TES, concerning the satisfaction of the cooling demand. Furthermore,
Figure 13 shows the three references of the relevant cooling powers, whereas
Figure 14 represents the
charge ratio of the TES tank in the proposed scheduling strategy. Moreover,
Figure 15 shows the temperature of the intermediate fluid, as well as the distribution of the temperature field of the PCM cylinders in the cooling, charging, and discharging processes.
Figure 12 shows that the refrigeration cycle without energy storage is not able to satisfy the cooling demand peak from
t = 5 h to
t = 8 h, thus incurring in some cooling power deficit, which is remarked in
Figure 12. However, the TES-backed-up refrigeration cycle operated according to the proposed scheduling strategy is shown to provide the required cooling power to the secondary fluid by combining the power produced at the evaporator and the TES tank contribution during the peak period. To do this, the TES tank is charged during the off-peak periods, as shown in
Figure 13b. The
charge ratio is shown to be kept within the safety limits, as depicted in
Figure 14, whereas a great deal of the energy capacity of the TES tank is used along the day. Notice in
Figure 15a that, during the charging processes, the temperature of the intermediate fluid is below
, being over
in the discharging process; in the stand-by processes, the thermal inertia of the intermediate fluid causes that, while approaching the thermal equilibrium with the PCM cylinders at
, the latter continues charging/discharging while the residual cold energy is transferred. The distribution of temperatures within the PCM cylinders represented in
Figure 15b show how the layers are quitting the latent zone from the outermost layer to the innermost, both in charging and discharging processes, as their latent energy depletes.
The performance of the low-level cooling power controller is not analysed in this work for the sake of brevity, since it is analysed in depth in [
25], thus no plots of the actual control actions are included in this work. In any case, some important issues concerning this control layer will be remarked upon. The settling time of the cooling power closed loop turns out to be small enough to assume, given the scheduling time, that the required cooling powers are almost immediately provided, as long as the computed references are feasible, which is ensured by the power constraints met by the scheduler. The hypothesis about the separation between the time scales considered in the modelling stage is confirmed in the aforementioned work. Concerning the charging and discharging cooling powers, constant cold energy supply/release is achieved by increasing the mass flow of the refrigerant/secondary fluid as the thermal resistance caused by the cylindrical shell in sensible zone grows. Further details about the performance of the cooling power controller can be found in [
25].
Regarding economic cost, the overall energy costs along the day of both the proposed scheduling strategy for the TES-backed-up cycle and the original system are gathered in
Table 5, as well as the achieved reduction. Notice that, in the case of the refrigeration cycle without storage, it is assumed that the energy deficit shown in
Figure 12 must be bought at the market price detailed in
Figure 9.
Obviously, the achieved reduction on the operating cost would come at a certain price in terms of CAPEX (capital expenditure). The economic analysis performed in this work is not intended to be comprehensive, but to throw some light on the high economic potential of such energy storage systems, which not only can ensure the satisfaction of demanding cooling load profiles, but they can also lead to significant cost reduction when properly operated. However, the simulation results highlight the need for suitable predictive scheduling strategies to manage the stored cold energy within the latent zone and fit the cooling power limits of a given facility. Indeed, the latter have been shown to vary significantly when the energy storage system is added to the original refrigeration cycle, both in charging and discharging processes.
5.6. Sensitivity Analysis
Some simulations of the proposed scheduling strategy considering parametric uncertainty are presented in this subsection. Given the design of the TES tank described in
Section 2, minor uncertainty is expected to be devoted to geometry, constructive materials, working fluids, and their thermodynamic properties. Nevertheless, a simple heat transfer model is used to compute the cooling power transferred between the PCM cylinders and the intermediate fluid
, where a probably uncertain heat transfer coefficient computed by classical correlations for natural convection [
32] has been considered to compute the thermal resistance
in Equation (
3). Moreover, the assumption of homogeneous heat transfer between the PCM cylinders and the intermediate fluid could lead to some degree of uncertainty regarding the cooling power calculation and then the evolution of the temperature of the intermediate fluid
. In order to assess the robustness of the proposed scheduling strategy against this kind of uncertainty, the latent temperature
considered by the prediction model will be artificially modified. The latter is usually precisely known by design, but if an uncertain value was considered in the prediction model, it would strongly affect the calculation of both TES charging and discharging cooling powers, which in turn determine how the intermediate fluid evolves. Therefore, the consideration of some uncertainty in the latent temperature
is just a simplified way of inducing uncertainty in all cooling power calculation within the TES tank.
Figure 16 shows the references of the TES charging/discharging cooling powers
and
, when some uncertainty in
is considered. This temperature has been increased/decreased in 1 K from the nominal value in the prediction model. Given the variations of the intermediate temperature for the nominal case shown in
Figure 15a, the considered increment/decrement is high enough to cause significant uncertainty in the prediction model with respect to the accurate model used for the plant simulation. Furthermore,
Figure 17 compares the TES
charge ratio for the three cases.
It is shown in
Figure 17 that the proposed scheduling strategy provides very similar results for the references of the charging and discharging cooling powers, in spite of the significant uncertainty in the cooling power calculation and thus in the estimated
charge ratio. In the case of
, the references are identical, as shown in
Figure 16b, but some appreciable differences appear in
Figure 16a regarding the charging power. The latter variations are due to the greater closeness to the upper limit imposed on the
charge ratio in the case of the charging process, as represented in
Figure 17. Therefore, the proposed scheduling strategy presents a suitable degree of robustness to significant modelling errors, ensuring cooling demand is satisfied and that the limits of the latent zone are met even if significant errors are introduced in the cooling power calculation.
6. Conclusions and Future Work
In this work, cold-energy management of a novel setup including a thermal energy storage tank based on phase change materials that complements a refrigeration facility has been addressed. The design of the upgrading process undertaken on the refrigeration facility to include the storage tank has been summarily described, along with the main features of the storage unit. Concerning the modelling, from a very detailed dynamic model previously developed, a simplified model focused on the slower dynamics related to heat transfer within the storage tank has been proposed to act as the prediction model in the scheduling optimization.
A layered scheduling and control strategy has been proposed, where a non-linear predictive scheduler computes the references of the main cooling powers involved, whereas the low-level controller ensures the achievement of the required cooling powers. The predictive scheduling problem is posed as a non-linear optimization with constraints due to cooling demand satisfaction, power feasibility, and latency limits, while considering economic criteria in the objective function.
A case study has been analysed, where a challenging load forecast that requires scheduling of different operating modes must be satisfied. The proposed strategy has been shown to ensure the cooling demand satisfaction and meeting of constraints while reducing the daily operating cost by up to 28% when compared to the refrigeration cycle without TES. Moreover, the latter fails in satisfying the cooling load during the entire day. A sensitivity analysis of the proposed strategy has shown it to provide satisfactory performance even when significant uncertainty in the prediction model is considered.
As future work, the application of the proposed strategy to the experimental plant is scheduled to be performed as soon as the upgrading process undertaken on the facility is finished. Furthermore, the development of an economic NMPC strategy is devised [
33], as well as the stability analysis of the proposed NMPC-based scheduling strategy.