Determining the Power and Capacity of Electricity Storage in Cooperation with the Microgrid for the Implementation of the Price Arbitration Strategy of Industrial Enterprises Installation

Abstract

The growing worldwide costs of energy produced as a result of conventional fuel combustion, the limited capacity of the distribution grid, and the growing number of unstable installations based on renewable energy sources increase the need to implement systems of stabilization and regulate loads for end users. The battery energy storage system (BESS) that operates in the internal microgrid of an enterprise enables the management of the accumulated energy in any time zone of the day. Using a price arbitrage strategy with an electricity storage facility, we can reduce the cost of high electricity prices during peak demand periods. This study aims to determine the most effective method of setting up the capacity and electrical power of an energy storage system operating in a microgrid, in an enterprise to implement a price arbitration strategy. Such a method should include consideration of the characteristics of the demand profile of consumer systems, the charges related to electricity, and electricity storage costs. The proposed deterministic method is based on the use of a defined parameter, “marginal income elasticity”. In this study, the size of energy storage refers to the power and electric capacity of BESS that are used for the implementation of the price arbitrage strategy.

1. Introduction and Review of the Literature Related to the Optimal Power and Capacity of an Electric Energy Storage System

In recent years, the energy market has seen an increase in interest in electricity storage, resulting in the development of scientific research on various working conditions and the strategies for their operation. Numerous studies have presented reviews of energy storage technologies in terms of their applications in microgrids [1,2,3,4,5,6,7,8]. Researchers presented the main functionalities that can be implemented in microgrids, including the absorption of energy from renewable sources, improvements in the quality parameters of electricity, peak shaving strategies, and price arbitrage and time shifts [9,10]. One of the main goals of the research has been to develop a methodology to achieve the optimal parameters of energy storage from an economic point of view, taking into account the investment and operating costs and the technical and economical parameters of various technologies that have potential for broad usage [11,12,13,14,15].

Research on the battery energy storage system (BESS) that uses deterministic and stochastic methods to determine the cost effectiveness of storage technologies was presented in previous works [16,17,18]. Adopted models were analyzed, including the costs of individual BESS technologies, the degradation of the capacity over time, and the losses of capacity during the discharge readiness period. The application of the integrated model to define and select energy storage parameters was presented in previous works [19,20,21]. These works presented models that included the implementation of thermal, electrical, and aging processes, as well as various sources and parameters that characterized the production of electricity within the microgrid. Stochastic predictive models, using 24 h wind force forecasting to optimize the power and capacity of energy storage in microgrid systems, were proposed in [22].

The problem of selecting the power and capacity of energy storage to balance microgrids, based on local results with various integrated renewable energy sources and various types of energy storage, was studied [23,24,25,26]. These studies presented mathematical models of microgrid systems with sources such as photovoltaic panels and wind turbines. On the basis of actual data characterizing the demand for electric energy, a simulation of the microgrid operation was performed, depending on the variability of electric energy demand in an island system. The use of electricity storage systems to increase the share of energy generated from renewable sources was also considered in previous works [27,28,29,30].

The issue of selecting the size of energy storage for households with their own renewable energy production systems for time-shifting functionality was discussed in an earlier study [31]. As a result of that research, it was shown that group energy storage, compensating for the flow of energy transferred to the external grid, is more profitable than individual storage systems. Studies on the maximization of expected daily economic profit, obtained using the time-shifting strategy to postpone the production of renewable energy, were presented in [32].

Korpikiewicz [33] broadly presented the conditions required for the operation of autonomous energy storage to implement a price arbitrage strategy, i.e., the use of variable energy rates throughout the day to reduce energy demand in periods of high energy prices and increase demand in periods of low energy prices. Algorithms describing the logic of determining the BESS charging and discharging cycles to optimize the operation of the system have been presented with the basic technical and operational data of BESS, which were obtained in various energy storage technologies [34].

A very important problem that should be considered when installing BESS in enterprises is the safety of the system. Particular attention should be paid to fire hazards posed by lithium-ion batteries. Therefore, the safety of BESS is the subject of intensive studies conducted by scientists [35], engineering associations [36], territorial units [37], and manufacturers who implement their own fire protection concepts [38]. Despite intensive research, there is still a lack of effective and rapid methods that could be widely used.

In summarizing the literature review, it should be noted that there are several studies on the use of BESS to implement a price arbitrage strategy. Most of the works dedicated to price arbitration focus on separated systems, supporting the distribution network and operating autonomously with constant and fixed charge and discharge values [39,40,41]. The models of energy storage operation presented in the literature confirm that the operating profit resulting from the use of the storage facility for price arbitrage is proportional to the total storage capacity.

In the available research, there are several studies on the profitability of an energy storage system management strategy that take into account the constraints associated with the actual energy demand and power of microgrids in production plants. Restrictions resulting from legal regulations on billing for the production of energy in a given country or in real microgrid systems may cause the benefits of using storage systems to decrease non-linearly with an increase of BESS capacity and power.

This study aims to determine the most effective method for setting up the capacity and electrical power of an energy storage system operating in an enterprise’s microgrid to implement a price arbitration strategy. Our research considered the existing technical and cost limitations in real enterprises that lead to a decrease in the effectiveness of the implementation of a price arbitration strategy. This paper defines the indicators for assessing the effectiveness of this strategy, and on that basis, we propose a determination of the effective boundary for BESS size. The microgrid system of the enterprise is a separate power installation, created from the load devices, active energy storage, or generation of assets with a control-and-regulation system, that is capable of managing the energy and electric power balance within the enterprise, connected to the distribution system operator’s (DSO’s) network. In this study, we assumed that the microgrid system managed an electricity storage installation and industrial power load within selected companies that were connected to a medium voltage grid.

2. BESS Work Strategy, Characteristics of Companies Selected for Research, and the Chosen BESS Model

The paper analyzes the use of BESS in terms of representative functionality for the electricity market, that is, price arbitrage. Price arbitration is based on the use of daily differences in unit prices of electricity. The essence of this strategy is the storage of energy purchased from the external grid in the price valley and then unloading the battery storage to supply the microgrid loads at times when the unit energy prices are the highest. When examining the conditions of this BESS functionality, one should consider the electricity prices [PLN/kWh] based on the offers of trading companies in the competitive market and the variable rates [PLN/kWh] for the distribution services that are included in the tariff of the appropriate distribution system operator that is approved by the President of the Energy Regulatory Office (ERO). The second option is to consider electricity prices according to the rates of the Polish Power Exchange Stock Market.

For customers who are billed according to the tariffs of energy companies, price arbitrage may be applied by selecting multizone tariff groups. The most diversified prices are in the B23 tariff group. There are three time zones in this group: S1 is the morning peak, S2 is the afternoon peak, and S3 is the the rest of the day. At all hours of the day on Saturdays, Sundays, and public holidays, energy is billed in the S3 zone as the rest of the day. The distribution of hours according to UTC + 1 time (coordinated universal time + 1 h) in tariff group B23 is presented in

Table 1. Distribution of time zones in the B23 tariff group for individual months in UTC+1 time. Yellow background with 1 is zone S1; red background with 2 is zone S2; green background with 3 is zone S3.

	00:00	01:00	02:00	03:00	04:00	05:00	06:00	07:00	08:00	09:00	10:00	11:00	12:00	13:00	14:00	15:00	16:00	17:00	18:00	19:00	20:00	21:00	22:00	23:00
Jan	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3
Feb	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3
Mar	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3
Apr	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
May	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
Jun	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
Jul	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
Aug	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
Sep	3	3	3	3	3	3	1	1	1	1	1	1	1	3	3	3	3	3	2	2	2	3	3	3
Oct	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3
Nov	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3
Dec	3	3	3	3	3	3	3	1	1	1	1	1	1	3	3	3	2	2	2	2	2	3	3	3

. The multi-tariff time zones included in Table 1 are typical for Poland and are applied in the tariffs of the four largest distribution network operators in Poland.

In the case of enterprises in the B23 tariff group, the price arbitration strategy is based on avoiding the purchase of electricity from the external grid when the variable unit rates in PLN/kWh are the highest, according to variable fees during the afternoon peak in the S2 zone.

An important element of electricity billing that should be considered when applying the price arbitrage strategy is the capacity fee. In Poland, beginning on 1 January 2021 as part of the implementation of the capacity market, an additional component was introduced to settlements for distribution services: i.e., the capacity fee in PLN/kWh. The rate of the capacity fee is published annually by the President of the Energy Regulatory Office (ERO), along with the designated hours of peak power demand during the day, at which times this component should be added to the consumed kilowatt hours Because the amount of this fee depends on the hours of the day, it increases the daily difference in prices related to electricity consumption and affects the application of the price arbitration strategy [42].

The capacity fee in Poland, after 1 January 2021, applies to all enterprises and is charged from 07:00 to 22:00 on business days. The hours in which the capacity charge applies according to UTC + 1 are presented in

Table 2. Schedule of hours in UTC+1, time of charging the capacity fee valid from 2021; red background on 1.

	00:00	01:00	02:00	03:00	04:00	05:00	06:00	07:00	08:00	09:00	10:00	11:00	12:00	13:00	14:00	15:00	16:00	17:00	18:00	19:00	20:00	21:00	22:00	23:00
Jan	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
Feb	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
Mar	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
Apr	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
May	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
Jun	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
Jul	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
Aug	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
Sep	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0
Oct	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
Nov	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0
Dec	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0

.

The price arbitrage strategy research was carried out for three different companies on the basis of the time series of the average 15 min electric load power consumed by the companies and recorded by measuring systems in an annual period. Data recorded in individual 15 min intervals were marked as load power $P_{L15}$ [kW]. The companies selected as research subjects were marked with letters A, B, and C, to which the year of registration of the time series tested was added (2018 and 2019). The selected companies carry out production activities with the use of various technologies and in various specialties. The enterprises are powered from the medium voltage power grid. Companies A and C are characterized by a constant level of energy consumption on working days; their work is carried out in a three-shift system. Enterprise B works in two shifts, only on working days. The differences in the weekly work organization of the enterprises are visible in Figure 1, which presents the average weekly profiles of power demand in 15 min power-demand intervals.

The characteristics of the organization of the work in the surveyed enterprises and their power demands are also illustrated by the coefficients of variation in the statistics of the 15 min power consumption time series, as shown in

Table 3. Statistical data of the 15 min load power time series of the enterprises.

Enterprise Average 15 min Load Power in Year		A2018	A2019	B2018	B2019	C 2019
Maximum	kW	1822	1860	509	508	1272
Average	kW	1217	1177	208	202	786
Median	kW	1336	1361	228	205	786
Standard deviation	kW	353	434	175	174	176
Coefficient of variation		29%	37%	84%	86%	22%

.

Based on the regulations that govern the application of tariff rates by energy companies and the settlement rules on the energy market and the capacity market, simulations of the BESS effect for the “price arbitrage” functionality were carried out. As part of the research, analyses of the time series of parameters characterizing the operating state of BESS, which were created as a result of the simulation of its operation in the microgrid system and the size of settlement data at the point of common coupling (PCC), were carried out.

The research consisted of adopting subsequent parameters characterizing the size of the energy storage, increasing them by a fixed value, and simulating their operation for a “price arbitrage” strategy in 15 min intervals for the entire annual measurement period. To investigate the price arbitrage strategy related to electricity, the input was the increasing capacity in kWh. The results of successive “k” simulations at the given BESS capacities were the quantities that described the effects of BESS.

In the case of microgrids, price arbitrage may be carried out by charging the energy storage from the power grid operated by one of the DSOs in periods when the cost of electricity from internal microgrid sources is lowest. Energy storage is discharged through the receiving systems in periods when the cost of electricity from the DSO grid is highest. As part of the price arbitrage implemented in the microgrid, financial benefits are obtained by taking advantage of the price difference between the avoided purchase of energy from the grid during discharging and the price of energy supplied by BESS. Typical BESS operating states for the implementation of the price arbitrage strategy are presented in Figure 2.

The revenue obtained resulting from the use of BESS for price arbitration, REV_BA, for one charging and discharging cycle, results from the use of energy stored in the E_BA energy storage for the company’s needs, taking into account the depth of discharge planned for the price arbitration, DoD_A, and the maximum price, C_ESmax, of energy not taken from the DSO grid in a given zone S, as presented by Equation (1).

$$RE{V}_{BA}=E_{BA}·(1-Do{D}_A)·C_{ESmax}$$

(1)

The operating costs of price arbitration with BESS are marked as O_A for one charge and discharge cycle. The costs include charges for energy collected during BESS charging from the DSO network in the S zone at the minimum price, C_ESmin, on a given day. The operating costs include the efficiency of the storage system, η_B, resulting from losses related to the conversion of AC/DC and DC/AC in the charge-and-discharge cycle. This cost is written as follows:

$$O_A={\frac{1}{\eta }}_B{E}_{BA}·(1-Do{D}_A)·C_{ESmin}$$

(2)

The operating income for a single cycle, $IN{C}_{BA}$, with a multi-zone tariff group, can be written as follows:

$$IN{C}_{BA}=RE{V}_{BA}-O_A=E_{BA}·(1-Do{D}_A)·\left(C_{ESmax}-{\frac{1}{\eta }}_B{C}_{ESmin}\right)$$

(3)

If BESS is used for price arbitrage in the microgrid system, based on the energy supplied by external suppliers from the DSO’s grid, the purchase of O_E electricity and the cost of providing the distribution service in the O_D variable part should be considered when calculating revenues and costs. For this reason, the income for a single discharge cycle for microgrids should be calculated by including the separate revenues and costs of electricity, i.e., REV_BE and O_E, and for the distribution service, i.e., REV_BD and O_D:

$$IN{C}_{BA}=[({\mathrm{REV}}_{BE}-O_E)+({\mathrm{REV}}_{BD}-O_D)],$$

(4)

For a company in the B23 tariff group, the income for a given billing period resulting from the use of the storage system in n_i cycles, assuming one cycle per day and considering the discharge in the peak zones S1 and S2 and the resale of surplus energy at market prices, C_Erk, to the DSO grid, is calculated according to the following relationships:

$$\begin{gathered} IN{C}_{BA}=n_i·E_{BA}\left\{\right[[(1-Do{D}_{S1})·C_{ES1}+(1-Do{D}_{S2})·C_{ES2}+(1-Do{D}_{SR})·C_{Erk}]-{\frac{1}{\eta }}_B· \\ (1-Do{D}_A)·C_{ES3}]+[(1-Do{D}_{S1})·(S_{ZS1}+S_{ZJ}+S_{Oze}+S_{kog}+S_{Pcap})+(1-Do{D}_{S2})· \\ (S_{ZS2}+S_{ZJ}+S_{Oze}+S_{kog}+S_{Pcap})-{\frac{1}{\eta }}_B·(1-Do{D}_A)·({\mathrm{S}}_{ZS3}+S_{ZJ}+S_{Oze}+S_{kog})\left]\right\}, \end{gathered}$$

(5)

The unit prices of electricity, C_ES1, C_ES2, C_ES3, and C_Erk, are the prices that are accepted for settlements from the offer of electricity trading companies. The unit variable rates for the distribution service, S_ZS1, S_ZS2, S_ZS3, S_ZJ, and S_Oze, S_kog, S_Pcap, are calculated or adopted by the territorially competent distribution system operators in the form of a tariff approved by the President of the Energy Regulatory Office.

2.1. Assumptions Made in the Simulation Model

The simulations were carried out for enterprises A and B based on data from 2018 and 2019 (the series were marked as A2018, A2019, and B2018, B 2019) and for enterprise C based on data from 2019 (the series was marked as C2019).

The following assumptions were made for the simulation model:

• The effectiveness of individual strategies was tested for a full one-year period.
• Time series and validity times of individual price components were compared with UTC + 1.
• The energy storage tested was a storage equipped with lithium-ion batteries, which resulted in the highest degree of commercialization for this type of battery [43,44,45].
• To ensure the comparability of the results for all simulations, regardless of the tariff group that is used in a selected company, the same electricity price rates and the same rates for the distribution service in enterprise A in 2021 were used (tariff group B23 together with a power fee).
• The average 15 min BESS charging power value, P_BC, could not exceed the contractual power, P_U, that was accepted for settlement with DSOs, considering the average 15 min load power of P_L15 loads. This condition for each 15 min interval is described as follows:

  $$P_{BC}\le P_U-P_{L15}$$

  (6)
  On this basis, a condition was formulated defining the maximum charging power for each of the compartments:

  $$max{E}_{BC}\le \frac{15\left[min\right]}{60\left[min\right]}·1\left[h\right]·\left(P_U-P_{L15}\right)$$

  (7)
• The contractual power was assumed as the highest capacity of all registered 15 min average capacities in the examined billing period. Although this value is unknown at the time an enterprise determines the contracted capacity for a given settlement period, adopting it at the lowest level and not causing additional costs of overruns constitutes the most restrictive limitation for the use of storage capacity for the price arbitration strategy.
• The storage tank was unloaded only in zone S2, as this zone possess the highest unit.
• The rate of the S_Pcap capacity fee was calculated in the daily hours of peak demand for power in the power system, in accordance with the rules established by the Energy Regulatory Office for 2021.
• The time zone for charging the energy storage, Z_C, was programmed in the hours in which the S3 zone was valid and the power fee was not applicable.
• The energy of the discharging storage system was limited to the energy consumed by the energy receivers during the period in which the S2 zone was valid. This limitation was aimed at eliminating the discharge of the BESS “onto the DSO grid”, i.e., the negative flows at the settlement point. This situation is unfavorable because electricity is sold back to the external network at prices lower than the avoided costs of its purchase.
• With the above assumptions and the condtion described by Equation (6), the income was calculated in accordence with the following relationship:

  $$\begin{array}{cc}\hfill IN{C}_{BA}=n_i·E_{BA}& [(1-Do{D}_{S2})·(C_{ES2}+S_{ZS2}+S_{ZJ}+S_{Oze}+S_{kog}+S_{Pcap})\hfill \\ & -\frac{1}{{\eta }_B}(1-Do{D}_A)·\left(C_{ES3}+S_{ZS3}+S_{ZJ}+S_{Oze}+S_{kog}\right)]\end{array}$$

  (8)
• Charging and unloading cycles occurred in the time zones in force for the B23 tariff group, excluding statutory non-working days and holidays designated by employers.
• As the conversion efficiency of the charge-discharge cycle, η_B, was included on the charge side, the need to modify the charging power was provided, inclusive of the power to cover the conversion losses. The efficiency of the conversion system was assumed in the calculations to be ${\eta }_B=$ 85%.
• The depth of discharge was assumed to be:

  $$DoD=Do{D}_A=Do{D}_{max}=20\%$$

  (9)
• CAPEX costs and the BESS life cycle were not considered in the study. The analyses were limited to the operational economic effects realized by the price arbitration in accordance with the rules of electricity billing law in Poland.

3. Simulations of the Effectiveness of Price Arbitration Implemented in Microgrid Systems with the Use of BESS

The use of price arbitrage in enterprises entails a complication in programming the BESS operation control system, resulting from the need to include the complex and unpredictable profile of electrical loads. The demand for energy and power in the microgrid varies over time and results from the current demand for electricity by devices, implementing production processes, building infrastructure, servicing of communication routes, transport, social needs, etc. Additionally, it should be noted that each enterprise has a different nature of organizational and technological processes, i.e., each enterprise has its own individual specificities in running a business, which are connected with the demand for energy needs at certain times.

3.1. Indicators of the Effective Selection of Storage Capacity for Price Arbitrage

To assess the use of various BESS values for the implementation of price arbitration strategies in the enterprises, simulations were carried out. On the basis of the simulations, the implementation of the strategies was assessed. We assumed from the input data that the capacity of the E_BA reservoir increases step-by-step by a constant value. Thus, defined parameters were used, which were determined for each tested capacity value based on the annual measurement results. The list of defined parameters is presented in

Table 4. Measures for the evaluation of the functioning of subsequent BESS figures for the implementation of the price arbitrage strategy.

Parameters	Description
BESS maximum discharge power [kW]	Maximum discharge power for all 15 min intervals. This power determines the current carrying capacity of the inverter in the DC/AC direction.
BESS maximum charge power [kW]	Maximum charging power for all 15 min intervals. This power determines the current carrying capacity of the inverter in the AC/DC direction.
Annual energy charged [kWh]	Total energy introduced to BESS during the year as a result of charging.
Annual energy discharged [kWh]	Total energy siphoned off from BESS during the year as a result of discharge in zone S2.
BESS capacity utilization [%]	Indicator describing the degree of utilization of the available storage capacity, determined according to the following formula:
	$\mathrm{BESS}\mathrm{capacity}\mathrm{utilization}=\frac{{{\displaystyle \sum }}_{tp=1}^{tp=35040}{E}_{BD,tp}}{Number of working days ·E_{BA}·(1-Do{D}_A)}$,	(10)
Annual income [PLN]	Annual REV_BA income calculated as the sum of the income for each 15 min period during the year.
Marginal income elasticity	The relative increase in income obtained by the relative increase in the BESS capacity for the next simulation “k”:
	$\mathrm{Marginal}\mathrm{income}\mathrm{elasticity}=\frac{\frac{\left(IN{C}_{BA,k}-IN{C}_{BA,k-1}\right)}{IN{C}_{BA,k}}}{\frac{\left(E_{BA,k}-E_{BA,k-1}\right)}{E_{BA,k}}}$	(11)

.

Along with the increase in BESS capacity, increasingly smaller increases in annual income were observed, which were calculated as the difference between annual revenues and annual OPEX costs. In the case of microgrids, revenue is limited not only by the size of the energy storage, but also by the amount of energy consumed by the load in the price zone in which the storage is discharged and by the maximum value of the contracted capacity. As a result, the revenues do not grow linearly as they do in the case of the classic standalone BESS operation for price arbitrage, but grow according to a curve with a decreasing slope and a linearly increasing energy storage capacity. The impact of the indicated limitations on the operation of BESS in the enterprise microgrid is illustrated by the graph in Figure 3 that shows the temporal variability of the amount of energy stored in BESS for two arbitrarily selected storage capacities, 1000 kWh and 4500 kWh, in one of the weeks characterized by the highest energy consumption by the A2019 enterprise:

With an energy capacity of the storage system of 1000 kWh, the charging and discharging cycles were evenly distributed on the working days from 11 March 2019 to 17 March 2019. There were no limitations to this capacity that made it impossible to charge the magazine to a given value. The exception was on 13 March 2019, when restrictions related to the enterprise’s microgrid resulted in incomplete recharging of the storage system to the value of 987 kWh, representing 99% of the total capacity. At the same time, virtually all energy stored in BESS (98% to 100% E_BA) was used for all cycles. For comparison, with an energy storage capacity of 4500 kWh, incomplete charging occurred every working day and the storage capacity was used only from 63% to 77% on these days $E_{BA}$.

The simulation data for one year, which was obtained using price arbitrage, together with the BESS capacity that increased in successive steps with a constant contracted capacity equal to 1860 kW, are presented in

Table 5. Results of simulations of the price arbitrage strategy for A2019 enterprise.

BESS Capacity (Pu 1860) [kWh]	BESS Maximum Discharge Power [kW]	BESS Maximum Charge Power [kW]	Annual Energy Charged by BESS [kWh]	Annual Energy Discharged by BESS [kWh]	BESS Capacity Utilization [%]	Annual Income [PLN]	Marginal Income Elasticity [%]
100	27	10	23,059	19,600	94%	4268
200	53	21	46,118	39,200	94%	8535	100%
300	80	31	69,176	58,800	94%	12,803	100%
400	107	42	92,235	78,400	94%	17,070	100%
500	133	52	115,293	97,999	94%	21,338	100%
600	160	63	138,349	117,597	94%	25,605	100%
700	187	73	161,405	137,194	94%	29,872	100%
800	213	84	184,461	156,792	94%	34,139	100%
900	240	94	207,518	176,390	94%	38,406	100%
1000	267	105	230,570	195,984	94%	42,672	100%
1100	293	115	253,606	215,565	94%	46,936	100%
1200	320	125	276,634	235,138	94%	51,198	100%
1300	347	136	299,645	254,698	94%	55,456	100%
1400	373	146	322,631	274,236	94%	59,711	100%
1500	400	157	345,551	293,718	94%	63,953	99%
1600	427	167	368,409	313,147	94%	68,183	99%
1700	453	178	391,219	332,536	94%	72,404	99%
1800	480	188	413,983	351,886	94%	76,618	99%
1900	507	199	436,700	371,195	94%	80,822	99%
2000	533	209	459,341	390,440	93%	85,012	99%
2100	560	220	481,894	409,610	93%	89,186	98%
2200	587	230	504,363	428,709	93%	93,344	98%
2300	613	241	526,725	447,716	93%	97,483	98%
2400	640	251	548,973	466,627	93%	101,601	97%
2500	667	261	571,084	485,422	93%	105,693	97%
2600	693	272	593,022	504,069	93%	109,753	96%
2700	720	282	614,733	522,523	93%	113,771	95%
2800	747	293	636,166	540,741	92%	117,738	94%
2900	773	303	657,336	558,736	92%	121,656	93%
3000	800	314	678,221	576,488	92%	125,521	92%
3100	827	324	698,778	593,961	92%	129,326	91%
3200	853	335	718,983	611,136	91%	133,065	90%

.

The research showed that among the proposed indicators for the use of the price arbitration strategy, the parameter of marginal income elasticity was characterized by the greatest volatility. This is illustrated in Figure 4.

In Figure 4a, the characteristic point can be determined, beyond which the character of the curve changes from moderately sloping to a curve with a significant decrease. This point, defined by the authors as the characteristic point of the curve, determines the value of the BESS capacity, above which its further increase is ineffective. Figure 4a shows that the BESS capacity utilization waveform (green) is a less indicative parameter in determining the optimal BESS capacity, as there is no clear characteristic point on the curve. Even more difficult is identifying the “characteristic point”, which shows the non-linearly decreasing efficiency with increasing energy storage capacity, that is caused by the annual income parameter, as presented in Figure 4b.

Figure 4a,b shows that after exceeding the characteristic point, the parameter value of the marginal income elasticity begins to decrease significantly, along with the constant increase in the capacity of BESS. For the same capacity increases, the parameters of BESS capacity utilization (Figure 4a) and annual income (Figure 4b) are more linear. On the basis of the simulations, a conclusion can be drawn that the effective operation is increasing the capacity of BESS to the value for which the internal limitations of the microgrids do not have a significant impact on the effect of the implementation of the price arbitrage strategy.

In this study, we arbitrarily assumed that the effective value of the BESS capacity is determined when the parameter of marginal income elasticity is equal to 95%. From the results presented in Table 5 for enterprise A2019, the marginal income elasticity of 95% was achieved with the BESS capacity equal to 2700 kWh. In the marginal income elasticity diagram shown in Figure 4, this point is located before the characteristic point. The BESS capacity of 2700 kWh can be considered as the effective size of the energy storage capacity of enterprise A2019. This capacity value corresponds to the maximum charging and discharging powers in the 15 min intervals during the year, considering the work of BESS for price arbitration and the implemented technical limitations. The maximum values of these powers, as presented in Table 5, were calculated as a result of the simulation for the BESS = 2700 kWh capacity. There was also a minimum power size of the inverters for the assumed BESS capacity, as follows:

• A discharge power corresponding to DC/AC conversion 720 kW;
• A charging power corresponding to AC/DC conversion 282 kW.

The differences between the maximum charging and discharging powers are due to the fact that the charging period is 9 h and the discharging period is 5 h to 3 h, depending on the period of the year. Therefore, it follows that the discharging current is significantly higher than the charging current.

3.2. Validation of Indicators Based on Data Obtained from Enterprises (B 2019, C 2019 and A2018)

To verify the method of determining the optimal BESS dimensions for the implementation of the price arbitration strategy using the marginal income elasticity parameter, the B2019 and C2019 time periods were tested in a manner analogous to the method described for the A2019 enterprise. The results are shown in Figure 5.

In enterprises A and B, only the marginal income elasticity parameter indicates the existence of a characteristic point that influences the effectiveness of the price arbitrage strategy, as shown in Figure 5. However, it can be observed in the parameter of marginal income elasticity in enterprise B that the characteristic point was more difficult to identify than it was in enterprises A and C. This difference was due to the different organization of the working hours in these enterprises. In enterprise B, on the last shift of the working day, the volume of electricity demand in the afternoon peak hours of the S2 zone decreased significantly. In these hours, which were already at low values of BESS capacity, there were cases of incomplete discharges of BESS capacity in the S2 zone. Thus, the decrease in the demand on the microgrid for electricity in the discharge zone, together with the limitation assumed in point 10 in Section 2.1, resulted in the ineffective use of price arbitrage.

The arbitrarily adopted value of 95% for the marginal income elasticity parameter clearly indicated the existence of an effective value of the energy storage capacity. The effective value of energy storage capacity was visible in the B2019 and C2019 graphs near or before the characteristic point that resulted from the limitations of the microgrid.

Table 6. Numerical results of the price arbitrage simulation for enterprises B2019 and C2018.

Enterprise	BESS Capacity [kWh]	BESS Maximum Discharge Power [kW]	BESS Maximum Charge Power [kW]	Annual Energy Charged by BESS [kWh]	Annual Energy Discharged by BESS [kWh]	BESS Capacity Utilization [%]	Annual Income [PLN]	Marginal Income Elasticity [%]
B2019	630	168	66	134,628	114,434	87%	24,916	95%
C2019	1820	485	190	422,822	360,531	95%	78,906	95%

shows the BESS parameters in points for which the marginal income elasticity parameter is equal to 95%.

It should be noted that the characteristic point in the case of enterprises B and C occurred for various parameters that characterized the use of storage capacity; for the parameter of utilization of the BESS capacity, it was approximately 87% for the series B2019 and 95% for the series C2019. In enterprise A, the value of this parameter was 93%. The simulations showed that the parameter of using the storage capacity (i.e., the utilization of the BESS capacity), did not change significantly, as evidenced by its flattened characteristics. For these reasons, it can be concluded that this parameter is not very useful in determining the value the of effective use of BESS for a price arbitrage strategy.

The indicator of optimal BESS selection for the same enterprise was also analyzed in relation to the consumption profile from the previous year. The calculation results for the data series A2018 and B2018 are shown in Figure 6.

The results from the simulations again indicated that the marginal income elasticity parameter remained the most sensitive. The remaining parameters, which quantified the size of the storage system for the use of the price arbitrage strategy, did not clearly indicate the existence of the characteristic point that could be used to determine the optimal size of BESS.

By assuming a marginal income elasticity value of 95%, we determined the effective storage size for the arbitrage price strategy. For data series A2018, this was a BESS capacity of 2600 kWh, which is close to the ‘characteristic point”. For the B2018 data series, this was a BESS capacity of 600 kWh located ahead of the characteristic point. A comparison of the results for 2018 and 2019 for enterprises A and B is presented in

Table 7. Numerical results of the price arbitrage simulation for the A2018 enterprise.

Enterprise	BESS Capacity (Pu 1822) [kWh]	BESS Maximum Discharge Power [kW]	BESS Maximum Charge Power [kW]	Annual Energy Charged by BESS [kWh]	Annual Energy Discharged by BESS [kWh]	BESS Capacity Utilization [%]	Annual Income [PLN]	Marginal Income Elasticity [%]
A2018	2600	693	272	604,700	513,995	95%	111,914	95%
A2019	2700	720	282	614,733	522,523	93%	113,771	95%
B2018	600	160	63	129,186	109,808	88%	23,909	95%
B2019	630	168	66	134,628	114,434	87%	24,916	95%

.

The data in Table 7 show that the individual parameters in 2018 and 2019 were similar. This means that each enterprise maintained its basic nature of demand in 15 min intervals in subsequent years. However, it can be seen that in the case of the same enterprise, the marginal income elasticity equal to 95% indicated a higher value of the optimal BESS capacity in 2019. Studies of the load profiles of the same enterprise for 2018 and 2019 showed an increase in optimal storage capacity by only one step of the set capacity in the calculations. This slight difference may be due to the different number of non-working days in the analyzed years, together with the associated Saturdays and Sundays.

For the data series A2018 and C2018, we also examined how the proposed indicator to evaluate the efficiency of selecting the size of the energy storage, defined as the marginal income elasticity, behaved for various contractual powers. Figure 7 shows the results of the simulation of the BESS operation, in accordance with the price arbitration strategy for the data series A2018 and B2018, during which the contractual power was increased stepwise by a constant value.

In previous studies, it was assumed that the contractual power was equal to the maximum power of all 15 min power consumptions in an annual period. This was a hypothetical value and, in fact, it was impossible to determine if there were no tools for actively lowering the consumed power. Adopting a certain level of contractual power determines the operation of the energy storage in the event of the implementation of the price arbitration strategy. The higher the contractual power, the greater the possibility of increasing the charging power.

For the A2018 data series, with the increase of the contractual power, the point of marginal income elasticity equal to 95% as a function of the storage capacity shifts to the right. This means that the contractual power had a significant impact on the effective use of the storage capacity. In the examined enterprise A, the increase in the contractual power resulted in a linear increase in the effective storage capacity, as shown in Figure 8a. Unlike data series A2018, the marginal income elasticity curves obtained for enterprise B (data series B2018) showed a weak dependence on the change in contractual power, as shown in Figure 8b). For increasing values of the contractual power, the obtained values of the ratio were the same or increased slightly.

These differences can be explained by the fact that the limitation of the BESS charging current depends not only on the contractual power, but also on the energy consumption profile during the charging zone hours. Both enterprises, A2018 and B2018, had different work organizations and differed in the level of energy consumption in the adopted Z_C charging zones. Enterprise A maintained a constant high level of energy consumption during the Z_C zone hours, and the energy consumption in the Z_C zone of enterprise B was significantly lower due to the two-shift work organization. It can be assumed that in the case of enterprise B, it was important to limit the amount of energy discharged by BESS to a value not greater than the energy resulting from the demand of internal consumers in zone S2, and that the limitation resulting from the contracted amount of contractual power was insignificant.

4. Discussion and Conclusions

Studies of real microgrid systems have shown that the nature and variability of electricity consumption by enterprises limit the effective use of price arbitrage strategies. These limitations, which are caused by the rules for billing for electricity and the instantaneous amount of energy load, determine the possibility of charging and discharging the storage system. As a result, the effectiveness of implementing the price arbitrage strategy decreases non-linearly with an increase in the BESS capacity, despite the programming of constant values of charging and discharging energies.

The limitations of the real microgrid systems mean that, for certain BESS capacity values, further increases in the energy storage capacity for the implementation of price arbitrage cease to be effective. To determine this value, the marginal income elasticity indicator was used. The curve of this parameter as a function of increasing BESS capacity has a characteristic point, after which the curve begins to significantly decline. Our research showed that the characteristic point appears near the value of the marginal income elasticity parameter, which is equal to 95%. Our research results showed that the application of the characteristic point of the marginal income elasticity curve to determine the size of the energy storage capacity establishes the limit of the BESS capacity, which is effective in implementing price arbitrage.

The determination of the effective size of energy storage, based on the marginal income elasticity parameter equal to 95%, will indicate the sizes of the effective storage capacity for the same enterprise in the following years. However, in these cases, one should consider the variability in energy and power demand caused by different numbers of days off work, as well as Saturdays and Sundays.

The effective use of energy storage capacity can be influenced by the value of the contractual power reported for settlements to DSOs, especially for enterprises with a continuous nature of production where the intensity of electricity demand does not decrease during BESS charging hours. In enterprises where production is not continuous and the organization of work occurs in one or two shifts, the amount of electricity demand of microgrid loads in the adopted period of energy discharge by BESS is of great importance for the effectiveness of price arbitrage. In cases where this demand is much lower than the maximum load value, the limitation resulting from the amount of contracted power is insignificant and the importance of limiting the amount of discharging energy of the storage system to the amount of energy that is consumed by the microgrid loads increases.

This study undertook simulations aimed at determining the power and capacity of BESS for the functionality of price arbitration. Our research had certain limitations, as outlined below.

• The legal regulations and all of the prices mentioned in this research are only applicable in Poland.
• This paper did not attempt to implement dynamic tariffs based on hourly SPOT market prices on the electricity exchange. A market game based on the difference in hourly electricity prices may turn out to be more effective than an alternative based on the B23 tariff group, and may constitute an important premise for further research.
• This paper adopted the capacity fee rules applicable in 2021. Our study did not analyze the various legally permitted rules for power charges in Poland or the method of calculating power fee reductions depending on the daily power profile, which were introduced in settlements from 1 October 2021.

The following future work is intended:

• Research on the possibility of obtaining synergy via the simultaneous use of price arbitrage strategy and strategy peak shaving. These functionalities are representative of two separate markets, i.e., price arbitrage in the electricity market, which is the domain of trading companies, and the peak shaving strategy, which covers activities in the capacity market, a consideration that is important from the point of view of distribution and transmission system operators.
• Further research is recommended to verify the fit of the Gaussian probability distribution to the deviations of the profiles in relation to the mean value. In addition, further research is recommended for the purpose of analyzing seasonal and cyclical data.
• In the field of price arbitration, further research is recommended to identify more precisely the characteristic points on the curves that are indicated in this paper, including the marginal income elasticity curve.
• In future work, it is recommended that the analyses be extended to include the CAPEX costs of BESS installations and their life cycles for various electricity storage technologies.