*3.2. Outputs*

The storage efficiency of the system was optimized from both the mechanical and the thermal points of view. For the CAES system, the final purpose is electrical generation; the efficiency of mechanical energy storage (*ηms*) is defined as the ratio between the electrical energy generated during the discharge phase (*Le*,*exp*) and the electrical energy spent on the compressor during the charging phase (*Le*,*med*).

$$
\eta\_{\text{lms}} = \frac{L\_{\text{ef,exp}}}{L\_{\text{c,mcd}}} \tag{1}
$$

For the HTTES system, the final purpose is the recovery of the thermal waste from the compressor, for which the coefficient of performance (COP) in calorific storage is defined as the ratio between the thermal energy stored in the system (*Qheating*) and the net electrical energy spent for storage " *Le*,*med* − *Le*,*exp*# .

$$\text{COP} = \frac{Q\_{\text{heating}}}{L\_{c,med} - L\_{c,exp}} \tag{2}$$

For the LTTES system, the final purpose is the recovery of cooling capacity, so the energy efficiency ratio (EER) in refrigerated storage is defined as the ratio between the cooling capacity stored in the system (*Qcooling*) and the net electrical energy spent to produce it " *Le*,*med* − *Le*,*exp*# .

$$\text{EER} = \frac{Q\_{cooling}}{L\_{\text{c,med}} - L\_{\text{c,exp}}} \tag{3}$$

For all three systems, the maximum electrical power consumption (*Pe*,*med*) was constrained, as shown in Table 3.

$$P\_{c,mcd} = \frac{P\_{\max}}{\eta\_{\mathcal{S}} \* \eta\_{c}} \tag{4}$$

**Output Variable Objective Type** *Pe*,*max* <3 kW *ηms* Maximize *COP* Maximize *EER* Maximize

**Table 3.** Output variables and optimization choices in the workflow.

For the CAES, the discharge time (*td*) was maximized.

$$t\_d = \frac{M\_{acc}}{G\_{exp}}\tag{5}$$

where *Macc* is the mass stored during the charging process and *Gexp* represents the mass flow rate of expanded air processed by the turbine [29].

*td* Maximize

The study of the optimization of the micro-CAES + TES system is not limited to simply maximizing the accumulation efficiency parameters; it is also necessary to combine other requirements that arise from the installation of such a system within a home unit, as follows:


## **4. Results**

The results show that of the 300 experiments generated, less than 34% meet the constraints imposed by the workflow. Unacceptable solutions were discarded from the design space table.

Figures 2 and 3 provide an estimation of the relationships between input and output factors. A positive value (red bar) indicates a direct relationship, while a negative value (blue bar) indicates an inverse relationship. In particular, Figure 2 represents the correlation between the maximum power absorbed by the compressor *Pe*,*max* and the input variables (*Vs*, *βmax*, *tc*), while Figure 3 shows the correlation between the CAES discharge time (*td*) and the input variables.

**Figure 2.** Relationship between the maximum power absorbed by the compressor *Pe*,*max* and the input variables (*Vs*, *βmax*, *tc*).

**Figure 3.** Relationship between the discharge time (*td*) and the input variables (*Vs*, *βmax*, *tc*).

To present the relationship between the mechanical energy storage efficiency (*ηnm*) and the input variables, a 4D bubble chart (Figure 4) was used. This is particularly useful because it allows the values obtained from optimization for the four variables to be shown. In particular, the x and y axes represent *βmax* and *ηnm*, respectively, while the diameter indicates the *tc* and the color indicates the *Vs*.

**Figure 4.** The x and y axes of the 4D bubble chart represent *βmax* and *ηnm*, respectively, while the diameter indicates the *tc* and the color indicates the *Vs*.

The correlation matrix (Figure 5) summarizes all of the relationships concerning the mechanical accumulation system between the input factors (*VS*, *tc*, *βmax*) and the output variables (*Pe*,*max*, *td*, *ηms*).


**Figure 5.** Correlation matrix to summarize the results referring to CAES. Values close to +1 indicate that the two variables are positively correlated, while values close to −1 indicate a negative correlation. If the value is close to 0, the variables are not correlated.

The results were also plotted with reference to TES—specifically, the 3D bubble chart in Figure 6 shows the relationships between COP, EER, and *βmax*.

**Figure 6.** The x and y axes of the 3D bubble chart represent COP and EER, respectively, while color indicates the values of *βmax*.

Figures 7 and 8 show the parallel coordinate charts of the COP and EER output variables, respectively, in the optimized case. This is a particularly useful tool for plotting the input and output variables of each design simultaneously.

**Figure 7.** Parallel coordinate chart of the COP output variable in the optimized case.

**Figure 8.** Parallel coordinate chart of the EER output variable in the optimized case.

#### **5. Discussions**

This section presents a discussion of the optimization process. As shown in Figure 2, the expanded charge times (*tc*) reduce the power absorption to the compressor. Instead, there is a direct relationship between the power absorption to the compressor and the storage volumes (*Vs*) and compression ratios (*βmax*). Figure 3 indicates that the CAES discharge time (*td*) is directly related to all three input factors. The most influential factor is the volume of the CAES tank (*Vs*). It is beneficial to have the longest possible discharge times (*td*) in order to have a functional mechanical storage system for the power supply of the residential building.

Furthermore, increasing the storage volume (*Vs*) can be useful within certain limits, as it can be seen that as this factor increases, the power consumption to the compressor also increases significantly. Increasing the charging time (*tc*) to the upper limit of 5 h, on the other hand, has a twofold positive effect: a reduction in power consumption and an increase in the discharge interval (the second most important factor after the storage volume).

It was assumed that the consumption of the building taken as a case study is concentrated in the evening hours, so during the day it accumulates electrical energy in elastic form for 5 h, and in the evening this energy is reconverted into electrical form to supply domestic users.

The maximum compression ratio (*βmax*) is almost irrelevant to the discharge time, but it is the second most important factor in terms of incidence with respect to the increase in compressor power. Therefore, within appropriate limits related to the space of the building and the purchase costs of the tanks, it is convenient to work with larger storage rather than higher storage pressures.

In Figure 4, it is possible to notice the inverse relation between *ηnm* and *βmax*. Therefore, in order to maximize the ratio between electrical energy produced during the discharge phase of the CAES system and electrical energy spent for the charge, it is necessary to reduce the compression ratio to a minimum. The reduction in the maximum pressure under which the CAES operates is a need in line with the considerations made previously for the other output variables. The efficiency of mechanical energy storage is substantially independent of the other input factors; it is not by chance that no trend emerges due to the diameter or color of the bubbles. With fixed *βmax*, there is the same efficiency for any combination of the factors *Vs* and *tc*, provided that the relative experiments have not been discarded from the design table. For this reason, in the low-yield zone of the graph and, thus, at the high pressures of the CAES, only a few combinations of the two factors are possible (blue bubbles with large diameters, which indicate low storage volumes with extended charge times). The high-yield zone is the one to focus on for further analysis. This zone is also the one that offers the most choice in the configuration of the optimal setup

for the system (bubbles of different colors with diameters of various sizes, meaning wider ranges of values in which to try to combine the input factors).

The resulting optimized CAES configuration, consistent with the interval thresholds set in the workflow of the problem, provides the following input factors for the solution:


Figure 6 highlights the fact that COP and EER are inversely proportional to the compression ratio. Therefore, for HTTES and LTTES systems, it is also convenient to work with *βmax* = 10.

To maximize COP and EER, a pair of *VS* and *tc* values is not specifically required, but it is essential that the compression ratio is set to a minimum. Therefore, the configuration adopted to optimize the CAES problem is also suitable for HTTES and LLTES systems. The fact that the trends of mechanical and thermal storage systems are not in contrast with one another is a very positive aspect. The input factor values of the experiment optimized for CAES can be confirmed for HTTES and LLTES systems as well.

The qualitative analysis of the trends allows us to define the optimized configuration for the micro-CAES + TES system under study.

In accordance with the equations reported in [29], starting from CAES, a single-stage reciprocating compressor (*N* = 1) is required to process an airflow rate of *Gmin* = 11.88 kg/h ≈ 2.75 L/s (FAD) to a maximum pressure of 10 bar. The compressor requires a 1.5 kW electric motor. A comparison of the results with real models from manufacturers' catalogues was carried out, and the reliability of the study model was confirmed. The compressor takes 5 h to fill the CAES tank from the initial pressure of 5 bar to the final pressure of 10 bar, consuming about 6 kWh of electricity supplied by the solar photovoltaic system in the process. The compressed air storage tank has a volume of 10 m3, equivalent to an overall length of 5.2 m by 1.65 m in diameter. During the loading phase, a total of almost 60 kg of air is stored to be used later for expansion.

Since the compressor is single-stage (*N* = 1), the HTTES system recovers the thermal waste from the charging phase exclusively through a heat exchanger located downstream of the compressor. The maximum temperature of the compressed air leaving the stage is estimated at about 190 ◦C, and the total heat recoverable in one charge is 8.6 MJ.

Moving on to the discharge phase of the CAES, this involves emptying the tank in order to supply the previously selected pneumatic reed valve motor, in order to deliver an electrical power of 3 kW to the user. The motor is supplied at a constant pressure of 5 bar, thanks to a pressure reducer installed downstream of the tank. Once this pressure threshold is reached in storage, the reducer interrupts the flow, and a mass of air equal to that discharged remains trapped in the CAES tank, i.e., about 60 kg. The discharge lasts a total of 0.17 h (≈10 min). The minimum air temperature at the end of the expansion is just below −66 ◦C, and the cooling capacity that can be recovered through the heat exchanger of the LTTES system is about 4.6 MJ.

The mechanical and thermal storage values used for the optimized configuration return *ηms* of 8.4%, COP of 0.43, and EER of 0.23.

#### *Preliminary Analysis of Costs*

Finally, a preliminary investigation of the system is presented, with the analysis of costs and relative comparison with the most common electrochemical storage systems on a small scale. Table 4 shows the purchase costs of the main components of the micro-CAES + TES system studied.


**Table 4.** The purchase costs of the main components of the micro-CAES + TES system.

To make the comparison with battery storage systems, it was considered that the average household consumption in the evening hours for a household amounts to approximately 4.5 kWh. Thus, for batteries, the cost analysis is summarized in Table 5.

**Table 5.** The purchase costs of storage batteries.


For lead–acid and lead-gel batteries, it should be noted that about 50% of energy is retained in the charge–discharge cycle to avoid damaging them. For this, a 9 kWh battery is used. For lithium-ion technology, on the other hand, 80% of use is considered to be 5.7 kWh. In terms of purchase cost, the micro-CAES system is also cheaper than batteries.

Given that the CAES system under consideration can generate 0.51 kWh of electrical energy to the user at a constant power of 3 kW, it was determined what configuration the CAES system should have in order to generate a significant amount of energy, taking into account the purchase cost and available space.

To obtain 3 kWh from the discharge, it would be necessary for the discharge to last for 1 h. At this point, there are two options: increase the maximum compression pressure, which compromises the already low efficiency of mechanical energy storage; or increase the storage tank size.

In the first case, a compressor with a maximum air pressure of 35 bar and a 13 kW motor is required, which is incompatible with the nominal power of a typical domestic photovoltaic system. In addition, the mechanical energy storage efficiency would drop to 6.16%.

In the second case, a 60 m<sup>3</sup> tank would need to be filled to 10 bar in 5 h using a 9 kW compressor, and the efficiency (a function of *βmax*) would remain fixed at 8.4%.

The efficiency of CAES (8.4%) is almost one-tenth of the efficiency of the most efficient batteries present on the market (70–90%).

The only advantages that a micro-CAES + TES system such as the one studied can offer compared to state-of-the-art batteries lie in its longer service life and the possibility of recovering the thermal waste to be used for heating/cooling the building. The years of service life for the system studied, as shown in Table 6, can be estimated by taking into account the average life of the compressor, which is the component that is potentially the most prone to failures and which, moreover, requires more attention for maintenance.

**Table 6.** Comparison of estimated service life years for different storage systems.


#### **6. Conclusions**

This study investigated the feasibility of compressed-air energy storage (CAES) systems, with the objective of carrying out the optimization of mechanical and thermal storage systems for small-scale trigeneration, designed for a single-family residential building equipped with a photovoltaic system with a rated power of 3 kW.

When dealing with a multi-objective optimization problem, the optimal solution obtained is never unique. The results identify a set of potentially optimal solutions. Among these, it is up to the decision-maker to choose which objective to favor in the optimization strategy. In particular, for this case study, the best configuration for the CAES was obtained under the following conditions:


The results of the optimization showed some critical issues. A very small percentage of the electrical energy stored in elastic form is again convertible into electrical energy. The efficiency of CAES is nearly one-tenth of that of the most efficient batteries on the market, and the discharge times are also extremely short. At the same time, for TES systems, only 43% of the net electrical energy expended is recoverable as heat in the HTTES system, with the hot source (compressed air) at 190 ◦C; in the LTTES system, the recoverable cooling capacity reaches about 23% in relation to the net electrical energy expended, with the cold source (expanded air) at a temperature of <sup>−</sup><sup>66</sup> ◦C. A tank with a volume of 10 m3 is not easy to install in a residential building.

The micro-CAES + TES system studied is inefficient, and is more expensive than batteries. The only advantages that micro-CAES + TES systems offer compared to batteries lie in their longer service life and in the possibility of recovering the thermal waste to be used for heating/cooling the building. The critical aspect of the system, optimized as much as possible in its configuration, can be attributed to the disproportion between the air consumption of the reed motor in the discharge phase and the flow rate that the compressor is able to process during the charging phase. The use of air motors, as widely accessible devices, has proven to be unsustainable for a micro-CAES system, even if optimized.

Therefore, in the future, this micro-CAES + TES system for small-scale trigeneration could be improved as a result of the optimization carried out in this work. If experimentation with new expanders is able to reduce the consumption of compressed air during discharge, the system could be configured as a viable alternative to batteries for the accumulation of photovoltaic energy. This is because the system offers indisputable advantages linked, as seen, to the exploitation of the thermal waste of the process to be used to heat/cool the building. Furthermore, the estimated useful life is double or even triple compared to some types of batteries currently on the market.

An interesting implementation of this study could involve the possibility of coupling the proposed system with an air–ground heat exchanger.

**Author Contributions:** Conceptualization, P.M.C., C.B., S.P. and D.M.; Data curation, P.M.C., C.B., S.P., D.M. and N.M.; Formal analysis, P.M.C., C.B., S.P., D.M. and N.M.; Investigation, P.M.C., C.B., S.P., D.M. and N.M.; Methodology, P.M.C., C.B., S.P. and D.M.; Resources, P.M.C., C.B., S.P. and D.M.; Software, P.M.C., C.B., S.P. and D.M.; Supervision, P.M.C., C.B., S.P. and D.M.; Validation, P.M.C., C.B., S.P. and D.M.; Visualization, P.M.C., C.B., S.P., D.M. and N.M.; Writing—original draft, P.M.C., C.B., S.P. and D.M.; Writing—review & editing, P.M.C., C.B., S.P. and D.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We thank Francesco Tramonte of the Department of Engineering for Innovation at the University of Salento for support in the development of this study.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**

