*Article* **Experimental Analysis of Residential Photovoltaic (PV) and Electric Vehicle (EV) Systems in Terms of Annual Energy Utilization**

**Wojciech Cieslik <sup>1</sup> , Filip Szwajca <sup>1</sup> , Wojciech Golimowski 2,\* and Andrew Berger <sup>3</sup>**


**Abstract:** Electrification of powertrain systems offers numerous advantages in the global trend in vehicular applications. A wide range of energy sources and zero-emission propulsion in the tank to wheel significantly add to electric vehicles' (EV) attractiveness. This paper presents analyses of the energy balance between micro-photovoltaic (PV) installation and small electric vehicle in real conditions. It is based on monitoring PV panel's energy production and car electricity consumption. The methodology included energy data from real household PV installation (the most common renewable energy source in Poland), electric vehicle energy consumption during real driving conditions, and drivetrain operating parameters, all collected over a period of one year by indirect measuring. A correlation between energy produced by the micro-PV installation and small electric car energy consumption was described. In the Winter, small electric car energy consumption amounted to 14.9 kWh per 100 km and was 14% greater than summer, based on test requirements of real driving conditions. The 4.48 kW PV installation located in Pozna ´n produced 4101 kWh energy in 258 days. The calculation indicated 1406 kWh energy was available for EV charging after household electricity consumption subtraction. The zero-emission daily distance analysis was done by the simplified method.

**Keywords:** energy consumption; real driving conditions; electric vehicle; solar panels; energy flow; renewable energy

#### **1. Introduction**

The introduction of environmentally friendly and highly efficient energy conversion systems represents one of the biggest challenges in the development of vehicle drivetrain systems [1]. Considering global energy consumption, 44% of global transport energy is consumed by light-duty vehicles, the next 26% by heavy-duty vehicles [2], and 30% by others [3]. Over 99.8% of transport means are still powered by combustion engines, impeding fast transport decarbonization [3,4].

The mechanical energy generated by petroleum product's combustion processes produces problematic carbon dioxide and other toxic exhaust compounds [5,6]. The EU legislation pays special attention to CO2 emission and, starting in 2021, has limited it to 95 g/km [7,8]. The CO2 emission is tightly bound with fuel consumption and related directly to engine efficiency in tank to wheel (TTW) calculations. A wide variety of powertrains systems, including internal combustion engines (ICE) and electric vehicles (EV), dedicated to vehicles demand not only tank to wheel analysis but also life cycle assessment (LCA). It provides detailed information about the environment's energetic impacts [9].

**Citation:** Cieslik, W.; Szwajca, F.; Golimowski, W.; Berger, A. Experimental Analysis of Residential Photovoltaic (PV) and Electric Vehicle (EV) Systems in Terms of Annual Energy Utilization. *Energies* **2021**, *14*, 1085. https://doi.org/10.3390/ en14041085

Academic Editor: Anastasios Dounis

Received: 5 January 2021 Accepted: 15 February 2021 Published: 19 February 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

LCA analysis carried out by Message et al. [10] shows the most significant climate change, expressed in gram CO2/km, for conventional vehicles using fossil fuels, particularly Petrol Euro 4, and the lowest for EVs, highlighting the significance of energy source. The elementary difference in LCA between conventional and electric powertrain relies on Well-to-Tank (WTT) and TTW share. EV characterizes the majority of the WTT energy conversion share compared to conventional ICE, where TTW is dominant [10]. Respectively, other studies indicate EV lifetime relevance [11]. The attractiveness of EVs in terms of global warming potential increases with its lifetime.

An electric vehicle's advantage is that it consumes energy from various energy sources, such as fossil fuels, renewable energy (RES), bioenergy, or nuclear [12]. The trends in EV technology field development provide a necessity for energy market analysis. The study presented by Xu et al., 2020 [13] discusses four different charging strategies, and how they will influence greenhouse gas (GHG) emissions in 2050, i.e., when high decarbonization levels and large RES shares are expected. The analysis included electricity mix type and reduced energy consumption from gas-fired power stations, and increased energy share from RES by controlling the charging process. Other predictions, up to 2040, indicate the benefits of implementing the ClimPol scenario. It significantly increases energy sharing from RES, reducing kilogram carbon dioxide equivalence per kWh to just above 0.2, from the global case of slightly more than 0.8 [14]. The trend is primarily influenced by an increased share of wind, solar biomass, and nuclear power in energy generation and EV battery charging, with results depending on the EU country [15]. Overall, the lowest charging effect has been achieved in France in 2015. Forecasting shows a 19% increasing RES share in 28 EU countries, along with a 17% decrease of share of electricity from solid fuels between 2020–2050.

Rising demand for RES energy led to fast photovoltaic infrastructure development, which became competitive in terms of low cost and high efficiency. The flexibility of the design of PV systems allows energy production in a wide range of voltage, from systems with power above 100 MWp to household applications, most often below 15 kWp [16,17]. Integration of household PV systems and electric vehicle use are a promising solution for global GHG reduction and locally lower fuel costs [17,18]. Energy analysis from Kyoto, with limited areas intended for RES, indicated 74% CO2 emission and 37% cost reduction from the power and transport sectors by applying photovoltaic rooftop systems and electric vehicles [19]. Results from 12 stands at an EV charging station equipped with photovoltaic panels (48 kW) and Li-ion 100 kW battery energy storage show the possibility of achieving 100% renewable electricity using appropriate control modes [20]. Modeling [21] of 400 combinations shows an attractive solution: cooperation of stationary battery (EV) with household PV infrastructure. The electric vehicle, being mobile energy storage, can effectively replace traditional battery storage. Using the battery in an EV as energy storage in such vehicle to grid (V2G) combinations, the self-sufficiency of solar self-consumption of household residentials increases [22]. In relation to V2G Technology, Wu Y. et al. proposes a real-time energy management system (EMS), that allows for a 29–55% reduction of the total cost of a Photovoltaic assisted charging station [23].

Energy stored in lithium-ion batteries has many advantages [24]. However, significant limitations are the dependence of the distance range on charging infrastructures, battery capacity, and drive quality [25,26]. Hence, it is important to monitor the high voltage battery state of charge to better understand energy flow phenomena and distance range prediction [27]. The range–distance prediction can be estimated based on various data collection methods. Zhang J. et al. obtained data from fifty EV taxis driving in Beijing [28]. In another study, testing was carried out by 32 electric busses traveling four routes under different working conditions [29]. Many studies about energy consumption prediction used data from experimental tests to validate new approaches [30–32]. In order to determine energy consumption by an electric vehicle, some researchers [33,34] used the real driving emission (RDE) test procedure as a suitable method to compare results with conventional ICE vehicles.

In recent years, Poland has noticed a rapid growth in interest in photovoltaic energy development [35]. The installed generated power increased by approximately 30% between the end of 2019 and May 2020 and achieved more than 1950 MW. The high growth rate places Poland in the top five EU countries in terms of new power. Most solar energy is produced by PV micro installations, representing more than 70% of all Polish PV markets in 2019. New regulations and supporting programs cause changes in the Polish renewable energy market [36]. The number of new registered battery electric vehicles has grown parallel to the number of household PV installations. Within a year, Poland's quantity of EVs increased by 80% to almost 7300 vehicles [37]. In effect, electric energy consumption will constantly be rising [38].

In the analysis of electric vehicle research, some major fields of experimental study can be quoted:


Regarding the high rate of changes in the energy market and non-ICE vehicle development, the authors decided to investigate the energy balance between energy production from the micro photovoltaic system and the energy consumption by small passenger cars equipped with a battery electric powertrain system. The approach of combining real objects (vehicle, residential building) in the energy balance consideration, as proven in the literature, is a commonly analyzed issue. In terms of the studied field, the novelty is the connection between the following approaches: annual balance, changing of ambient conditions, and road approved electric vehicle energy consumption evaluation based on a real driving conditions (RDC) test.

This study's aim is to assess household micro-photovoltaic systems' self-sufficiency in connection with battery electric vehicle use. The research goals are:


The extent of this study included the energy flow analysis from a PV system, then charging, followed by RDC testing to discover the amount of energy consumed by an electric vehicle. It covered a one-year period with energy generation analysis from photovoltaic panels being carried out during the same periods as the vehicle tests, with average energy consumption during driving estimated for both winter and summer measurements.

#### **2. Methods of Analysis**

The aim of this study was to evaluate the energy flow generated by the solar panels, followed by analysis of the energy flow during vehicle's charging and the energy consumption in the RDC test. The analysis of energy generation from photovoltaic panels was carried out during the same periods as the vehicle tests, i.e., in winter and summer periods in 2020 on the territory of Poznan city in Poland. In the same periods, the average driving energy consumption was also estimated. The following questions were posed: How much energy does a city electric vehicle consume during its intended operation periods? Is the applied 4.48 kW photovoltaic installation able to guarantee an electric vehicle's energy demand in the assumed driving scenarios? The scope of the research included two test runs compliant with the RDC test procedure in urban, rural, and motorway cycles in winter and summer conditions of 2020. The measurements were made in the ECO driving mode, which in earlier studies [34] showed a beneficial reduction in energy consumption by the vehicle. The measurements made in the same driving mode allowed the estimation of the impact of weather conditions on the overall energy consumption of the vehicle on selected road sections. The data from the test run were recorded in real-time based on the information pulled from the vehicle controller area network (CAN) by a dedicated

on-board diagnostics system (OBD) scan tool. The main parameters that were recorded during the measurements in the actual traffic conditions of the vehicle were those describing the operation of the electric motor (rotational speed, value of the current and voltage, and torque and vehicle speed) and the parameters concerning the accumulation of energy in the battery (state of charge (SOC), and power).

The characteristics of the electric vehicle used in road tests are shown in Table 1. The vehicle used in the tests—ŠKODA CITIGO<sup>e</sup> iV—is supplied with an electric drive, allowing different driving modes and variable intensity of regenerative braking. The 61 kW ŠKODA CITIGO<sup>e</sup> iV powertrain used a Li-Ion battery of 36.8 kWh full capacity and 32.3 kWh useable capacity. The location of the batteries have also been shown in Table 1.


**Table 1.** Technical data of the analyzed powertrain fitted in ŠKODA CITIGOe iV [34].

This article presents an analysis of three stages of energy conversion with emphasis on the vehicle's energy consumption under real traffic conditions (Figure 1). The comparison of the stages of energy conversion from the PV energy generation to vehicle charging, allowed us to develop a compilation of the possibility of driving the vehicle using energy from renewable sources only.

**Figure 1.** Division of analysis carried out in the discussed studies.

#### *2.1. Long-term Analysis of Solar Panel Household Power Generation*

Long term analysis of producing electric energy has been based on monitoring working the parameters of household micro-PV installation located in the northern part of Poznan and in service since February 2020. The system has been working in on-grid mode with solar electric power used by household and the surplus sold to the grid. The PV installation contains fourteen monocrystalline rooftop PV panels, DC/AC inverter, fuses, and assembly parts. The 320 W panels' total power output is 4.48 kW. The installation also includes three-phase 5.2 kW inverter SMA Sunny Tripower 5.0 and a two-way energy meter. The real-time view of the energy data (±1 Wh) is possible through dedicated web and mobile apps.

The daily average energy consumption has been calculated using Equation (1) below.

$$E\_{\text{household}\_{\text{usage}}} avg = \frac{\left(E\_{\text{from the grid}} + E\_{\text{direct use from solar panels}}\right)}{\sum \text{analyzed days}}\tag{1}$$

$$E\_{\text{household}\_{\text{usage}}} avg = \frac{2293 + 402}{258} = 10.4 \left[\frac{kWh}{day}\right]$$

The calculation was based on 258 days with 2293 kWh of energy taken from the grid and 402 kWh of energy supplied by PV installation.

Available energy dedicated to an electric vehicle charging has been estimated using Equation (2).

$$\begin{array}{l} \text{E}\_{\text{available for EV vehicle charging}} \\ = \text{E}\_{\text{from solar panels}} \\ - \left( \text{E}\_{\text{from the grid}} + \text{E}\_{\text{direct use from solar panels}} \right) \end{array} \tag{2}$$

$$E\_{\text{available for EV vehicle charging}} = 4101 - (2293 + 402) = 1406 \text{ [kWh]}$$

This additional energy of 1406 kWh is available for EV charging (taking into account the billing period of 258 days). Due to the analyses carried out in different periods of the year, it is possible to change the flow of energy from the source in the form of photovoltaic panels and the power plant. The trending differences are shown in Figure 2.

**Figure 2.** Characteristic correlations in electricity production from photovoltaic panels in the summer and winter months (based on [45]).

#### *2.2. Electrical Vehicle Charging Modes*

Small passenger EV used in this investigation could be charged following the modes below:


• 40.0 kW (AC/DC) combined charging system (CCS) from the rapid charging station. The combined charging system (CSS) can, within one hour, charge up to 80% of the car battery capacity.

The full charge time increases with decreased charging power. Charging profile of the first mode—2.3 kW (AC)—was observed over one full cycle. The charging level (±0.1%), voltage (±1 V), and current (±0.01 A) were sampled with 1.3 Hz and were registered from vehicle's CAN in real time. The observation setup is shown below in Figure 3.

**Figure 3.** Vehicle charging setup.

### *2.3. Vehicle Energy Consumption in RDC Test*

The test route was proposed in [34,46] and determined to lead through the city of Pozna ´n and its surrounding areas. It covered urban, rural, and motorway conditions. The maximum motorway legal speed is 140 km/h. Selected test requirements related to the course of the test run have been presented in Table 2. The duration of all the test runs exceeded 90 min, and the total length of the track did not change.

**Table 2.** Real driving conditions test requirements with map of the route traveled during the measurements [34].


The main problem of constantly developing industry is its negative impact on the environment. One of the most dynamically changing sectors of industry is transport, which significantly affects the concentration of hazardous substances in the air. In order to reduce the impact of vehicles on the environment, increasingly restrictive emission standards are being introduced, and solutions are being sought to minimize the emission of exhaust fumes from vehicles. Exhaust emission standards are set to control the pollutants emitted from automotive vehicles around the world. Exhaust emission values are measured under conditions in an established type approval test. This part of the vehicle certification process is responsible for the environmental performance of the vehicle and is the same for all passenger cars. The course of the test corresponds to the most likely road conditions, and the tests performed, which are the same for all vehicles, authorize the comparison of emission results between them. However, currently, more and more attention is paid to road tests, i.e., tests performed in real driving conditions. At present, these tests have been included in the European Union regulations under the name RDE (real driving emissions) [47,48]. They are performed in order to best reflect the actual vehicle operation conditions in terms of ecological aspects. Such tests must be performed with specific requirements, the main assumptions of which are presented in Table 2. The winter and summer runs performed in this research met the requirements specified in the RDE test directive (Table 3). However, due to the lack of exhaust emission measurement, these tests are named RDC.

**Table 3.** Meeting real driving emissions (RDE) test requirements for summer and winter performed measurements [47,48].


<sup>1</sup> and 2—Specific value determined for each trip, taking into account vehicle speed parameters based on European Union regulations. <sup>1</sup> Data calculated for winter real driving conditions (RDC) test. <sup>2</sup> Data calculated for summer RDC test.

#### *2.4. Energy Supply and Demand for Selected Driving Scenarios*

The choice of a means of transport for many users is motivated by economic factors (total cost of vehicle use). Literature sources of daily commutes provided average distances in the EU. Commutes within cities ranged between 4–25 km. Commutes from suburban areas can be significantly longer (Figure 4). In the next part of this section, we analyzed and presented three scenarios of daily commute.

Based on described trends [50] of dependency of distance and travel frequency and information about average distances covered in Poland [49], three different distances have been selected to analyze (Figure 5). Scenario 1 assumes a 15 km distance per day focusing on an urban area. Consequently, scenarios 2 and 3 concern the suburban areas where the residence is a farther distance from the workplace.

**Figure 4.** Average travel distances (km) in Europe, with focus on Stockholm and Barcelona, showing the dependence of the average distance on the distance from the metropolitan area [49].

**Figure 5.** Typical distribution of frequency of trips vs. distance covered in Italy [50], supplemented by proposed scenarios and statistical data on average distances covered in Poland [49].

#### **3. Results**

#### *3.1. Long Term Analysis of PV Energy Production*

In Poland, the solar radiation falls consistently within 1050–1160 kWh/m2/year, with highest values observed in the central part (Poznan or Warsaw) and in the south of the country (Krakow), as shown in Figure 6. The photovoltaic systems produce similar amounts of energy throughout the country. The Institute of Renewable Energy report shows that, in Poland, about 70% of currently installed photovoltaic sources are micro installations with an increasing trend. In 2019, there were 640 MW of new power installed, three times more than in 2018. Such a rapidly growing branch of power industry presents real possibilities for power self-sufficient households with excess energy to be dedicated to zero-emission transport.

**Figure 6.** Average annual sum of photovoltaic (PV) power potential [51] and the development of micro installations in Poland [35].

Measurements of an existing single-family building with a 4.48 kWp photovoltaic installation were made. The Sunny Tripower 5.0 model STP5.0–3AV–40 424 inverter enabled real-time measurement of the generated power by the photovoltaic system, with monitoring performed by a dedicated application (SMA Smart Connected) and data archiving by the Sunny Portal service. The inverter's parameters are shown in Table 4 and the values of the energy obtained are presented in Figure 7.

**Table 4.** Characteristics of the inverter Sunny Tripower 5.0 used in the tested home installation [52].


The differences between energy generation in the summer and winter months were significant. Shortening the time of solar radiation of the panels by 30% reduced the maximum power generated by 19% (Figure 7), resulting in a total reduction in the share of accumulated energy by about 40% (energy gain in June was 664.5 kWh while in March only 406.7 kWh). Such large differences in the total values of energy produced raises doubts about the ability to meet the energy demand for both the power supply to the building and the electric vehicle.

**Figure 7.** The data from 4.48 kWp solar energy installation located in Poznan city in Poland—energy generated during the winter and summer weeks, collected by the Sunny Tripower 5.0 data acquisition system.

The compiled daily electricity production characteristics (Figure 7) were averaged and presented in relation to the daily usage pattern of the electric vehicle (Figure 8). Part of the energy produced during the day (around the afternoon hours) was transferred to the power plant, because at this point, the energy was not used for the household's needs. The graph shows the maximum vehicle charging time with a discharged battery. Charging time, in this case, is long also because of the choice of basic charging technology. In the absence of energy production by photovoltaic panels, the energy needed to charge the vehicle (as well as other home usage) was drawn back from the power plant.

**Figure 8.** Electric vehicle charging scenario (using charging mode 2) in relation to the average generation of electricity from the photovoltaic installation (red—summer, blue—winter).

### *3.2. Vehicle Charging Analysis*

The charging profiles, represented by SOC, power, and voltage, are shown in Figure 9. The authors did not optimize the charging process. Algorithms implemented by the vehicle manufacturer controlled the charging. Battery charging from 15.2% to 95.6% lasted almost sixteen hours. The charging process was carried out by constant current (CC) mode, i.e., with approximate constant current and variable voltage using the original manufacturer household charger. The voltage rose from 296.25 V to 351.75 V at the end of charging. At the same time, the current value was changing significantly. At 96% of SOC, current reduced to slightly above 0, and with it, large voltage variations were observed. When the car was not being used and had a full battery, the charger ensured stand by energy consumption. The battery delivered 30.16 kWh energy after one charging cycle.

#### *3.3. The Impact of Atmospheric Conditions on the Energy Consumption of an Electric Vehicle*

The driving cycles, realized in compliance with the RDC procedures, were started at 100% battery SOC level (software readout). The test runs were performed by a single driver to assure consistency in the driving style. Compared to the previous analysis results of the driving energy consumption in the RDC test [34], this article focuses on the determination of the total energy consumption by route sections: urban, rural, and motorway under different atmospheric conditions. The vehicle velocity and relative SOC profile during the RDC tests are shown in Figure 10. Presented curves marked in blue and red colors represent different ambient conditions. Vehicle speed and state of charge are represented by solid and dotted lines, respectively. In the tests with similar conditions achieved during the measurement journeys, the indications of the vehicle speed in relation to the distance travelled showed a high similarity to the journey, both in the urban part and in the sections with increased speeds. Some differences in the speed on the suburban and highway routes were dictated by road conditions, and there was no possibility to repeat them.

**Figure 9.** Single cycle charging profile carried out by household dedicated converter in constant current (CC) mode.

The driving cycles, realized in compliance with the RDC procedures, were started at battery level SOC 100% (software readout). The test runs were performed by a single driver to avoid inconsistency in the driving style. During the test run, the vehicle speed and battery level were recorded. Due to running the vehicle in ECO mode, the maximum speed was limited by the drivetrain controller; both runs were comparable speeds in the given test intervals, and nevertheless, the energy consumption was about 11% more in the winter period. For this reason, further work identifies the intervals of route split and vehicle speed affecting the increased energy consumption.

**Figure 10.** Comparison of the real driving conditions (RDC) test for the performed test runs in winter and summer conditions.

The flow of the energy ΔE was determined based on the flow of current (IBAT) and voltage (UBAT) of the battery as a result of its discharge and regenerative braking charge during driving of the car:

$$
\Delta \mathbf{E}\_{\mathbf{i}} = \sum\_{t=0}^{t=t\_{\text{max}}} \mathbf{U}\_{\text{BAT}} \times \mathbf{I}\_{\text{BAT}} \,\mathrm{d}\mathbf{t} \tag{3}
$$

• discharging:

$$
\Delta \mathbf{E}\_{\text{dis}} = \sum\_{t=0}^{t=t\_{\text{max}}} \mathbf{U}\_{\text{BAT}} \times \mathbf{I}\_{\text{BAT}} \mathbf{d}t \text{ (when } \Delta \mathbf{E}\_{\text{i}} < 0\text{)}\tag{4}
$$

• energy recovery (regenerative braking):

$$\begin{aligned} \Delta \mathbf{E}\_{\text{reg}} &= \sum\_{\mathbf{t}=0}^{\mathbf{t}=\mathbf{t}\_{\text{max}}} \mathbf{U}\_{\text{BAT}} \times \mathbf{I}\_{\text{BAT}} \mathbf{dt} \\ \text{(when } \Delta \mathbf{E}\_{\text{i}} &> 0 \text{ and } \mathbf{M}\_{\text{reg}} < 0) \end{aligned} \tag{5}$$

In order to determine the individual electric powertrain operating conditions, road portions were specified where the system operated in these individual conditions. On this basis, the operating modes were divided into individual phases: driving, acceleration, standstill, and braking, during operation of the electric drive. The adopted criteria have been shown in Table 5 and Figure 11.

**Table 5.** Vehicle motion phase criteria.


**Figure 11.** Distribution of motion phases in winter and summer conditions.

The energy balance as a function of the type of the road is shown below (Figure 12). Regardless of the weather conditions, similar energy recovery values were recorded. However, due to more vehicle stops during winter measurements, higher energy recovery was achieved in both the urban and suburban parts during winter measurements. The greatest amount of energy was recovered in the urban cycle due to the high number of brakings, compared to the smooth traffic road portions.

**Figure 12.** Energy consumption balance in terms of road and weather conditions.

The specification of the assumptions used in the energy flow summation (Figure 12) are specified in Table 6. The assumptions for the urban, rural, and motorway segmentation of the route are consistent with the RDC test assumptions presented earlier (Tables 2 and 3). The value of power delivered or generated from/to the battery was recorded during the measurements, with the following assumptions, the summed energy flows presented earlier were determined. These assumptions also apply to the energy consumption totals in Figure 14.

**Table 6.** Vehicle motion phase criteria [34,35].


During the RDC test, the vehicle's energy consumption was dependent on the road conditions. Increased energy consumption of the vehicle was noticeable in the higher speed ranges. The bar charts of Figure 13 show the energy flow and the share of energy consumption for different speed ranges. The marked points represent the total energy flow of both recovered (green) and consumed (red) energy in a specific speed range. The largest amount of energy was recovered in the 20–50 km/h range. The amount of energy recovered in the urban speed range allowed us to increase the vehicle range. However, the energy consumption in each speed range was higher than the recovered energy. The highest energy consumption was recorded in the intervals of increased vehicle speed, both during summer and winter driving conditions; in the speed range 110–120 km/h, the vehicle consumes more than 4 kWh (the distance covered at this speed is almost 25% of the entire RDC test). The shares of each speed interval in the test indicate nearly identical driving conditions in the urban route speed range (0–60 km/h) and in the motorway route range (v > 90 km/h).

**Figure 13.** Energy flow characteristic and the share of energy consumption for different speed ranges in the RDC tests.

A summary of the vehicle's energy consumption during the RDC test for sections of the route (urban, rural, and motorway) is shown below in Figure 14 and Table 7 (1, 2 and 3). It compares both the energy consumption without recovery and the reduction of energy consumption after taking into account the recovery of energy from braking, which is shown in Table 7 (1 , 2 and 3 ). The graph shows the energy flow characteristics for both winter and summer conditions. The energy consumption was then calculated for 100 km of the sections under consideration (urban, rural, and motorway), thus obtaining the total energy consumption of the vehicle in winter conditions (14.9 kWh/100 km) and in summer conditions (13.1 kWh/100 km). Averaging the total energy consumption of a vehicle, without division into sections of the route and weather conditions, gives 14 kWh/100 km. The assumptions presented below are affected by some simplifications, but the paper is intended to undertake a preliminary analysis of the possibility of supplying an electric vehicle from a renewable source. Due to the variety of drivers and routes taken, these calculations are not applicable to every type of vehicle or every road with their own unique characteristics; nevertheless, the authors estimated the average energy consumption for a small urban vehicle, which was then compiled together with the assumed travel scenarios of the electric vehicle user.

**Figure 14.** Energy flow characteristics and the share of energy consumption for different speed intervals in the RDC tests.


The above average values, without division into atmospheric conditions, were used for the analysis of the electric vehicle driving scenarios and are presented below.

#### *3.4. Energy Supply and Demand for Selected Scenarios of Driving a Vehicle*

The analysis of the average distance travelled by passenger vehicles in 2.4 has been used to develop a theoretical list of three scenarios (Table 8) in which the distance, together with the share of individual route sections, is a variable. Scenario (S1)—the minimum analyzed commuting distance is 15 km and covers urban driving conditions only. Scenario (S2) assumes a one-way distance of 30 km with 15 km in urban and 15 km in suburban driving conditions. Scenario (S3) assumes the participation of all three sections of the route and is 45 km total in one direction. Increasing the distance makes it necessary to charge the vehicle more often. This frequency was estimated based on calculations of the distance of a particular route compared with the energy consumption of the vehicle presented in the previous sections. According to the investigation's assumptions about charging, the vehicle must ensure an adequate charge to cover the entire route planned during the day. An overview of the charging frequency is presented in Figure 15.


**Table 8.** Analyzed scenarios of distances covered by an electric vehicle.

**Figure 15.** Frequency of charging an electric vehicle in the presented scenarios depending on the distance covered by a user.

The summary of energy production to power an electric vehicle, Figure 16, shows the energy produced by the photovoltaic installations in the months from March to October, marked as green. The calculated energy consumption of the vehicle in all scenarios shows it can be met by the production of electricity from solar panels in all months except October. However, if we also consider the current average electricity consumption of household appliances, Scenario 3 is not possible in any month. In such cases, the energy to power the vehicle in scenario 3 would come directly from the power plant. The solution to reduce electricity costs in such cases should be increasing the number of photovoltaic panels.

**Figure 16.** Energy balance of the presented scenarios of driving an electric vehicle in relation to the energy produced and consumed by a household with a 4.48 kW photovoltaic installation.

#### **4. Discussion**

This paper focused on an experimental assessment of the energy flow between household PV installation and a small size battery electric vehicle. Karkosi ´nski et al., 2018 [53] conducted a similar approach without RDC test requirements and household energy consumption. The results indicated a significant impact of the sun's shining state during the day for energy generation, obtaining 100 W/m2 sun irradiance on a cloudy day and almost 250 W/m<sup>2</sup> on a clear day in January. In effect, 11.4 kWh energy was obtained on an average sunny day in January. The study of [54], with polycrystalline and monocrystalline PV modules, showed an 80% difference between the amount of energy produced in summer and winter over a three year period. The results of this study confirmed the above trends through detailed analyses of PV energy production during the week of ride test cycles. In addition, a 19% increase in peak power has been observed in summer with the effective energy production time during the day raised by approximately 4 h.

EV energy consumption was examined with RDC tests. The average energy consumption in considered cases was 13.1 kWh/100 km in summer and 14.9 kWh/100 km in winter. The elevated energy consumption was likely caused by interior comfort system energy demand. For reference, in [33] where energy consumption was determined also with the RDC test, the average energy consumption was 19.6 kWh/100 km, calculated from two repeated rides with the difference between the rides of 1.3 kWh.

Comparison of three types of powertrain: gasoline, hybrid, and electric, in terms of an RDC test, was performed by Pielecha J. et al., 2020 [55]. During the RDE test, the lowest accumulated energy demand was achieved for the electric vehicle, approximately 30% lower than the combustion engine and 10% lower than a plug-in hybrid powertrain. In the research presented, like in [55], energy consumption was analyzed separately for each road type and in terms of ambient climate conditions, not powertrain type. In [36], the influence of drive mode and braking strategy on energy flow in small-sized EV was investigated within the RDC test requirements.

During the summer, smaller energy consumption was observed, 17% for urban conditions and 22% for rural residents, including braking energy recovery. The result of another investigation [56] shows decrease of possible driving range from 150 km at 20 ◦C to 85 km at 0 ◦C. Doyle A. et al., 2019 [57] indicated that interior thermal comfort systems consume an average of 14% of the total trip's energy by the cooling system and 18% by a heating operation.

The last part of the investigation included analyses of energy balance between household PV installation and EV. To assess sufficiency of PV installations, three scenarios of distance covered by EV were considered. In effect, it was possible to use an EV car charged by surplus energy from PV installation. The mentioned aspect of cooperation between electric vehicles and renewable energy sources is key to effective electric powertrain future developments.

#### **5. Conclusions**

The article presents an analysis of energy flow from the stage of production of electricity from a renewable source in the form of solar energy (PV panels), through the charging of the electrical vehicle, and the subsequent consumption of this energy while driving. Charging of electric vehicles, especially in areas with limited access to charging points, can be difficult. Therefore, the estimates of both the energy consumption of the vehicle and the necessary frequency of charging the vehicle are shown here. The energy consumption of the vehicle has been recorded for driving conditions during both winter and summer periods. The influence of the type of route (urban, rural, or motorway) and distance covered have significant impact on the vehicle's energy consumption. The presented scenarios are a stage of preliminary elaboration by the authors of the mechanism of simulating the energy consumption of electric vehicles considering various road conditions. In addition, oversupply of energy produced by the residential PV system used in this study indicates the possibility of reliance of charging of the EV from that source only.

The study specific conclusions:

	- In winter conditions: 11.39 kWh/RDC test (estimated at 100 km = 14.9 kWh)
	- In summer conditions: 10.35 kWh/RDC test (estimated at 100 km = 13.1 kWh)
	- For all three scenarios in March–September period without energy demand by household appliances
	- For Scenario 1 and 2 in April–August period with household appliances

**Author Contributions:** Conceptualization, W.C., F.S., W.G., and A.B.; methodology, W.C., F.S., W.G., and A.B.; formal analysis, W.C., F.S., W.G., and A.B.; investigation, W.C. and F.S.; writing—original draft preparation, W.C., F.S., W.G., and A.B.; writing—review and editing, W.C., F.S., and W.G.; visualization, W.C. and F.S. 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:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** This work was supported by the Volkswagen Group Polska Sp. z o.o.

**Conflicts of Interest:** Authors declare no conflict of interest.

### **Abbreviations**


### **References**


## *Article* **Description of Acid Battery Operating Parameters**

**Józef Pszczółkowski**

Faculty of Mechanical Engineering, Military University of Technology, 00-908 Warsaw, Poland; jozef.pszczolkowski@wat.edu.pl

**Abstract:** In this paper, the operating principles of the acid battery and its features are discussed. The results of voltage tests containing the measurements conducted at the terminals of a loaded battery under constant load conditions, and dependent on time, are presented. The article depicts the principles of the development of electric models of acid batteries and their various descriptions. The principles for processing the results for the purpose of the determination and description of the battery model are characterized. The characteristics under stationary and non-stationary conditions are specified using glued functions and linear combinations of exponential functions, and the electrical parameters of the battery are determined as the components of the circuit, i.e., its electromotive force, resistance, and capacity. The dynamic characteristic of the battery in the form of transmittance was determined, using the Laplace transform. Possible uses of the crankshaft driving signals as diagnostic signals of the battery, electric starter, and internal combustion engine are also indicated.

**Keywords:** acid battery; battery operating parameters; testing and modeling

#### **1. Introduction**

The lead–acid battery is a chemical source of electric energy in which current is generated as a result of chemical processes taking place on its electrodes in the presence of sulfuric acid. The factor that forces the course of the current generating processes is an electromotive force of the cell resulting from a difference in the electrodes constituting the cell normal potentials. The basic parameters characterizing the electrical and energy properties of the battery are: voltage, twenty amp hour (Ah) rate capacity, and the ability to start an engine (CCA—Cold Cranking Amps). CCA is a rating used to define the ability of the battery to start an engine in a cold temperature. The existing chemical models of the battery explain a mechanism of the generation of an electromotive force and a sum of its electrical and energetic capacities, e.g., electric capacity. However, the chemical models are not useful for analyzing electrical circuits where the acid battery is a component. When using an acid battery, it is not possible to determine the current electrical parameters of the circuit, current, and voltage. In such a circuit it is necessary to use electric battery models composed of the typical electrical circuit components: electromotive force, resistance, capacitance, inductance, and others [1]. Modeling of the batteries, including acid batteries, has become necessary and is carried out in a particularly intensive manner due to the increased demand for electricity in vehicles resulting from the arrival of electric and hybrid drives. The modeling and determination of the battery model parameters is considered a difficult, unclear, laborious, expensive, and ambiguous process [2].

A lead–acid battery consists of a negative electrode made of porous lead and a positive electrode consisting of lead dioxide. Both electrodes are immersed in electrolyte which is a solution of sulfuric acid and water. The overall reversible chemical reaction, which enables lead–acid batteries to store energy, is as follows:

$$\text{PbO}\_2 + \text{Pb} + 2\text{H}\_2\text{SO}\_4 \Leftrightarrow 2\text{PbSO}\_4 + 2\text{H}\_2\text{O}$$

Discharging a battery causes the formation of lead sulphate at both the negative and positive electrodes. Sulphate from the sulfuric acid electrolyte surrounding the battery is

**Citation:** Pszczółkowski, J. Description of Acid Battery Operating Parameters. *Energies* **2021**, *14*, 7212. https://doi.org/10.3390/en14217212

Academic Editor: Wojciech Cieslik

Received: 12 October 2021 Accepted: 29 October 2021 Published: 2 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

used in the formation of this lead sulphate. When the battery is in the fully discharged state, its two electrodes are of the same material and there is no chemical potential or voltage between these two electrodes. Between the fully charged and discharged states, the lead–acid battery experiences a gradual reduction in voltage. A voltage level is used to indicate the state of charge of the battery. Thus, there is a dependence of the battery voltage on the battery state of charge. The battery is in equilibrium only in the state characterized by no load. The battery voltage and its capacity have specified values. The battery under load is not in equilibrium, and its voltage and battery capacity differ significantly from the equilibrium values. The difference between the voltage at equilibrium and that under a load, with a current flow, is termed the battery polarization.

The battery voltage value or its dependence on time versus battery operating condition parameters is the basic battery operating parameter or characteristic. The operating characteristics of the acid battery are the object of research and modeling for the implementation of many practical and theoretical objectives: the evaluation of the correlation of the starting parameters of the internal combustion engine [3]; the design of the internal combustion engine start-up systems [4]; the analysis of the dynamic properties of the battery in electric vehicles [5]; the possibility of using the characteristics in the process of diagnosing the internal combustion engine and its starting system [6]. In the case of the lead–acid battery model in electric or hybrid vehicles, the charging and discharging process is of great importance, i.e., a charging/discharging voltage and state of charge (SoC) [7]. Very often the model of the lead–acid battery for the Stop-Start Technology is a circuit model with two resistance–capacitance (RC) blocks [8]. The simulated battery operating parameters are the voltages, currents, and state of charge (SoC). The battery models for the different designs of the lead–acid-based batteries, i.e., batteries with gelled electrolyte and an Absorbent Glass Mat (AGM), differ from the common lead–acid batteries models in regards to the parameters of the battery model, although they are based on the same chemistry [9]. There are also different models of the lead–acid battery in terms of their ageing processes, i.e., deep discharge models which are combined with a sulfation model [10]. The ageing processes determine the battery state of health (SoH). The purpose of some works is to investigate factors which affect the failure of automotive batteries or battery durability. The main factors influencing the aging process of batteries are the battery temperature and the discharge current [11]. Statistical methods are used for analysis and prediction of battery degradation in electric vehicle use [12], including regression models for estimation of the battery state of health [13]. In regression models, charge/discharge cycle number, battery terminal voltage, and internal resistance are used as independent variables.

The existing methods of battery testing have been systematically developed, and new approaches are used to determine the characteristics of batteries, e.g., based on neural networks, genetic algorithms, or Kalman filters. These often concern the determination of model parameters, the battery state of charge, and energy management in energy storage systems using batteries. Genetic algorithms are used to optimize the energy system of electric vehicles because of the growing number of electricity consumers in the vehicle [14]. For the state of charge of batteries, and its dynamic determination, supervised chaos genetic algorithms have been used [15]. The use of the Kalman filter based on the *RC* model for estimation of model parameters and the state of charge of lead–acid batteries requires knowledge of the value of the process covariance and the measurement noise [16]. The prediction voltage and lifetime of a lead–acid battery may be determined using neural network methods [17]. Ref. [18] describes the design of a measurement system to conduct the electrical tests, and an estimation algorithm for automatic analyses and reporting proceedings for lead–acid started batteries. Determination of the state of charge (SoC) of a lead–acid battery was tested using the electrochemical impedance spectroscopy (EIS) method [19]. Lead–acid cells were explored during intermittent discharge and charge processes. More battery parameters were taken into account in the design and simulation

of a model of a lead–acid battery [20]. These parameters were the SoC, battery voltage, and temperature of the battery in the charge and discharge state.

Previous research has also investigated the problems of the physical phenomena that determine the operation of the energy storage system, i.e., the lead–acid battery [21]. A relatively similar new modeling method for lead–acid batteries combined the physicochemical model with the equivalent circuit model [22]. Ref. [23] drew attention to the fact that the battery equivalent circuit model has two time constants. Because the load process duration is often short, the test data during the load period may not contain sufficient information for extracting these time constants. In contrast, the relaxation period may last hours, and thus may provide sufficient data for this purpose. The problems of the modern technology used in the battery production process were considered in [24]. Carbon materials are widely used as an additive to the negative active mass and allow the battery specific energy and active mass utilization to be increased. A constant problem concerning the use of battery energy relates to the starting of an automobile engine in low or negative temperature conditions. When using an autonomous means of engine pre-heating, it is necessary to optimally distribute the battery energy to the pre-start and start-up discharges [25].

The objective of this paper is to present the author's mathematical models of the acid battery for stationary and non-stationary dynamic operating conditions. The basis for the development of the models was the research results of voltage measurements at the terminals of the loaded batteries under constant load conditions, i.e., the dependence of the voltage on time. The battery tests were carried out on a test stand that was placed in a low-temperature chamber, which allowed the ambient and tested battery temperatures to be changed.

In the literature, the accumulator battery models are presented in graphic or mathematical form, and a mathematical description is not frequently used. Therefore, in this work the mathematical form of the model is particularly emphasized and explained. On the basis of test results, a linear model of the dependence of the battery terminals' voltage on its nominal electric capacity, loaded current, temperature, and battery state of charge (SoC) is elaborated. This multidimensional model was developed for the stationary operating conditions within the time period of several seconds following switching on the load. In this case, the principles of planning the experiment were applied [26]. The dynamic characteristics of the battery are also presented, i.e., its voltage at the dynamic state of operating just after switching on the load, and after switching it off.

The principles of processing the results for the purpose of the determination and description of the battery models are characterized. The characteristics under the stationary and non-stationary conditions are specified using glued functions and linear combinations of exponential functions, and the electrical parameters of the battery are determined as the components of the circuit, i.e., its electromotive force, resistance, and capacity. Possible uses of the crankshaft driving signals as diagnostic signals of the battery, electric starter, and internal combustion engine are also indicated.

#### **2. Materials and Methods**

The battery performance tests were carried out on a test stand that was placed in a low-temperature chamber, which allowed the ambient and the tested battery temperatures to be changed. The equipment of the test bench enabled the test implementation and the recording of the battery operation parameters to be controlled. The tested battery was loaded with a constant resistance value within approx. 10 s. The values of the current and voltage were recorded by means of a computer measuring system, including after switching off the load, to observe changes in the electromotive force of the battery polarization during this period. The examples of the recorded dependencies of the current and voltage at the terminals within the load test of the battery of 54 Ah capacity are shown in Figures 1 and 2.

**Figure 1.** The current drawn from the battery.

**Figure 2.** The voltage (to 10 s) at the loaded and unloaded battery terminals.

At the moment of switching on the load, characteristic and correlated, proportional changes in the intensity of the absorbed current and voltage at the battery terminals can be observed. During the initial discharge period, the voltage at the battery terminals and the current decrease approximately exponentially, and then their values stabilize. When the load is switched off, the voltage increases rapidly and then increases exponentially (Figure 2). This is due to an increase in its electromotive force caused by changes in the electrolyte concentration in the vicinity and in the inner layers of the active mass of the battery plates.

The recorded characteristics have, in addition to the visible and clear trend of changes, significant irregularity. This can cause difficulties in their further processing to determine and interpret electrical characteristics and the battery model. Therefore, the courses were subjected to pre-processing aimed at smoothing them. The causes of signal distortion were analyzed, and methods of their elimination were developed. The following sources of interference were identified:


The various forms and principles of averaging were adopted as the methods of smoothing of the received signals. These can only be used in the case of a good recognition of the signal and an understanding of the nature of its changes, to ensure that useful signal components are not lost. The own noise of the measuring system, particularly high values, is usually represented as a single isolated deviation of its value from the set level. In principle, all distortions can be reduced using a method analogous to the moving average; the difference is that the moving average is a forecasting method in which the forecast

value is assumed as the moving average of the preceding values. In this case, a calculated value of the mean was taken as the central data value. It is preferred that this uses an odd number of datapoints. Depending on the degree of a signal interference, the smoothing of the curve can be used several times.

As a characteristic of the battery load, the dependence of the voltage at its terminals at a constant value resistance load is considered. The analysis concerns the characteristics during the battery load period, as shown in Figure 3 as Uload. After the load is turned off, the voltage at the battery terminals increases abruptly, and then gradually stabilizes to the value in the no-load state.

**Figure 3.** The voltage at the loaded (Uload) and unloaded (Uunload) battery terminals.

The different battery models are applied to the specific purposes and the different methods used to test the battery characteristics [27,28]. The simplest model of the battery presents it as an ideal voltage source, i.e., an electromotive force that does not exhibit even any internal resistance. The lead–acid battery is most often treated as a voltage source of electric current with a defined electromotive force and a variable internal resistance. In the electrical circuit, certain voltage changes at its terminals (at a constant value resistance load) can be justified by a change in its electromotive force or internal resistance. The changes in the electromotive force (or the internal resistance) are caused by the processes in the electrolyte around the electrodes or on their surface. When under the given discharge conditions, a constant value of electromotive force is accepted, and a classic equivalent electrical circuit of the battery can be presented, as shown in Figure 4.

**Figure 4.** The classic equivalent electrical circuit of the battery.

When such a battery is loaded with the external resistance R or the constant current I, the voltage at the battery and receiver terminals is as follows (1):

$$\mathbf{U} = \mathbf{R}\mathbf{I} = \mathbf{E}\_{\mathbf{B}} - \mathbf{I}\mathbf{R}\_{\mathrm{int}}.\tag{1}$$

A complex battery model takes into account its electromotive force, internal resistance, inductance or capacitance, and other characteristics. The model presented in this article considers the dependencies of the characteristics of the battery on its rated capacity, temperature, and state of charge. However, the model should reflect the principle of the lead–acid battery. It also should be simple, fast, and effective to implement and use. The equations of the lead–acid model always contain constants that must be determined experimentally by

laboratory tests. Any battery model can be validated by a simulation using, for example, the MATLAB/Simulink Software [29].

#### **3. Results**

#### *3.1. Stationary Operating Characteristics of an Acid Battery*

The operating voltage of the lead–acid battery depends on its rated capacity, current consumption, temperature, and state of charge (technical state). Previous research of oneand two-dimensional operating characteristics of acid batteries formulated a conclusion about the linear nature of the dependences of the battery operating voltage on the abovementioned independent parameters [30]. In order to determine the multidimensional characteristics of the acid battery operation, an experiment was developed that enabled determination of the coefficients of a linear equation that describes the relationship between the voltage of the loaded battery and the factors influencing it. Testing the characteristics of the batteries was carried out on a test stand prepared and placed in a low-temperature chamber, which enabled the operating conditions of the battery to be changed. In the initial period of load, changes in the current and voltage values are visible, resulting from the dynamic nature of the tested battery operation (load switching on). Because the objective of the study was the determination of the characteristics under the steady-state load conditions, the values of the parameters describing the operating state of the battery were determined for the load duration time of approx. 10 s, i.e., after their stabilization.

Gaining an understanding of the properties of the research object and its behavior under the influence of extortion requires many, often costly and time-consuming experiments. The number of measurements performed depends on the complexity of the model, the number of independent variables affecting the research object, and the variables' values. For the purpose of limiting the number of measurements and, at the same time, obtaining as much information as possible, it is necessary to plan the experiments and then perform them according to the principles resulting from the adopted plan. In the experimental research, the most commonly adopted approach is a linear structure of the model, in addition to exponential or logarithmic structures that can be reduced to a linear form. The method of least squares is most often used to determine the coefficients of these models. The method of least squares is used to identify the linear models and the second-order polynomials, in addition to the power or logarithmic functions.

The research object is characterized by the independent (input) variables *x*i, i.e., a set of parameters influencing its properties; and dependent variables *y*i, i.e., output quantities (the result of the interaction of input and disturbing quantities). The disturbing quantities *z*<sup>i</sup> are the result of an impact of random factors on the research object and the inaccuracy of measurement methods and means.

The research is carried out according to a prepared experiment plan, usually in accordance with the experiment table included in the plan. The determination of the inaccuracy of the measurement results is possible when the same experiment is repeated several times. The arithmetic mean can be used as a position measure and the standard deviation as a dispersion measure. The optimization of the model describing the real object mainly consists in finding the best of all possible limitations, and a model that describes the relationships between the studied variables.

As mentioned above, the experiment plan was developed with the assumption of a linear structure of the mathematical model that describes the relationship between the voltage of the loaded battery (dependent variable) and the physical quantities that influence the voltage (independent variables): battery nominal capacity—Q [Ah]; load current intensity—I [A]; temperature (of electrolyte)—T [◦C] (in this case it is more advisable and convenient to use the Celsius temperature scale than the Kelvin temperature scale); and battery condition—k. A two-level, static, determined, and complete experiment was assumed. The plan assumes that the input factors, i.e., the independent variables, take two levels of values: the upper ones are marked as "+1" and the lower ones are marked as

"−1". Therefore, the number of tests in the planned experiment for the four independent variables is: n = 2<sup>4</sup> = 16.

For individual independent variables, the appropriate symbols *x*<sup>i</sup> were adopted, and the levels of their variability were assumed. The levels of variability define the range within which they take the values [xmin − xmax]. To develop the model, it is necessary to code the input quantities. Coding consists of transforming the value of any input quantity into a coded (normalized) value which is within the range limited by the conventional levels of the input variables, and falling into the following set [−1 ÷ +1]. For this purpose, mathematical operations were performed, consisting of determining central values; that is, calculating arithmetic means for the individual variables and determining a unit of variation for the individual quantities, which is the unit value of the input factor change. These values were then coded. The units of the variables' variation were determined on the basis of the changes in the parameters of the operation of the engine starting system under the average engine starting conditions with the use of an electric starter.

A calculation of the units of variation consists of determining the unit value of the change in the independent variable. The unit of variation was determined on the basis of Equation (2). The value of ximax and ximin in Equation (2) corresponds to the maximum and minimum value, respectively, of the independent variable with the number, i.e., xi in the adopted variation range.

$$
\Delta \mathbf{x}\_{\mathbf{i}} = \frac{\mathbf{x}\_{\mathbf{i}\text{ max}} - \mathbf{x}\_{\mathbf{i}\text{ min}}}{2}. \tag{2}
$$

The central values are the arithmetic means of the maximum and minimum values of each individual independent variables Equation (3):

$$\mathbf{x}\_{\rm{io}} = \frac{\mathbf{x}\_{\rm{i}\,\max} + \mathbf{x}\_{\rm{i}\,\min}}{2}. \tag{3}$$

Coding of independent variables results in transforming the values of the input quantities into dimensionless numbers contained in the following set [−1; +1]. Coding makes the experiment plan independent of the real values and the physical meaning of independent variables describing the research object, and replaces the independent variables with dimensionless values. Hence, the methods of planning the experiment become universal and independent of the physical importance of factors describing a given phenomenon, and can be used in various fields of research.

Thus, the coded value of any independent variable, according to Equation (4), is:

$$\mathbf{x}\_{\rm ik} = \frac{\mathbf{x}\_{\rm i} - \mathbf{x}\_{\rm io}}{\Delta \mathbf{x}\_{\rm i}},\tag{4}$$

where the individual component of Equation (4) has the following meaning:


The appropriate levels of the variability of the factors were adopted, for which the values were, respectively:


The notional levels of the factor values are described as −1 for the lower value and +1 for the higher value. The first step of the experiment is to code the variables, which then assume conventional, dimensionless values. The central values of the independent variables were determined in the form of the arithmetic mean of the values assumed by these variables at the upper and lower levels according to Equation (3). Then, the units of

variability of the factors considered in the experiment were calculated. After calculating the central moments and units of variability, the variables were coded in accordance with Equation (4). The results of the activities leading to the presentation of the variables in the coded form are recorded in Table 1.


**Table 1.** Variables' coding results.

The dependent variable, i.e., the voltage at the terminals of the loaded battery, U [V], is also coded. Following the coding of the variables, the next step in the preparation of the experiment plan is the plan table arrangement, according to which the measurements are carried out. Table 2 presents the table for the planned experiment in question. The number of planned experiments results from the number of variables describing the research object and the number of levels of the values that these variables take.

**Table 2.** The matrix of the experiment.


The x0 value is an intercept of the linear model describing the object, and the subsequent columns represent independent variables, respectively: battery rated capacity, load current, electrolyte temperature, and battery condition (SoC—state of charge). A single experiment from Table 2 defines the measurement system as a set of independent variable values. Only one value of each variable belongs to each set, and all independent variables describing the research object were simultaneously taken into account. In the created plan, the number of experiments was 16. It is also assumed that the individual experiments included in Table 2 should be performed in a random order. The measurements were made for the tests presented in the table.

The linear regression equation describing the relationships between the variables for the presented plan takes the form of Equation (5):

$$\mathbf{y} = \mathbf{a}\_0 + \mathbf{a}\_1 \mathbf{x}\_1 + \mathbf{a}\_2 \mathbf{x}\_2 + \mathbf{a}\_3 \mathbf{x}\_3 + \mathbf{a}\_4 \mathbf{x}\_4. \tag{5}$$

The coefficients of the regression equation were determined using the following relationships Equations (6) and (7):

$$\mathbf{a}\_0 = \frac{1}{\mathbf{N}} \sum\_{i=1}^{\mathbf{N}} \mathbf{x}\_{0i} \mathbf{y}\_{\text{mean}} \tag{6}$$

$$\mathbf{a}\_{1\div4} = \frac{1}{\mathbf{N}} \sum\_{i=1}^{N} \mathbf{x}\_{1\div4i} \cdot \mathbf{y}\_{\text{mean}}.\tag{7}$$

After confirming the adequacy of the model, the equations were decoded and written in the form of a linear function taking into account all independent variables. The sought-after linear model of the object describes the relationships between the variables Equation (8).

$$\mathbf{y} = \mathbf{a}\_0 + \mathbf{a}\_{\mathbf{i}} \cdot \frac{\mathbf{x}\_{\mathbf{i}} - \mathbf{x}\_{\mathbf{i}0}}{\Delta \mathbf{x}\_{\mathbf{i}}} + \dots + \mathbf{a}\_{\mathbf{n}} \cdot \frac{\mathbf{x}\_{\mathbf{n}} - \mathbf{x}\_{\mathbf{n}0}}{\Delta \mathbf{x}\_{\mathbf{n}}}.\tag{8}$$

After carrying out the measurements according to the plan review table, the coefficients of the linear equation describing the relationships between the variables were determined according to Equations (6) and (7). After determining the coefficients, the following equation was obtained in a coded form Equation (9):

$$\mathbf{y} = 11.25 + 0.109\mathbf{x}\_{\mathbf{Q}} - 0.364\mathbf{x}\_{\mathbf{I}} + 0.135\mathbf{x}\_{\mathbf{T}} + 0.155\mathbf{x}\_{\mathbf{k}}.\tag{9}$$

This expression presents a linear mathematical model of the object, i.e., an acid battery, the coefficients of which, determined on the basis of the data from Table 2, indicate how much the value of the dependent variable (battery voltage) will change when the value of the coded independent variable changes by one.

The mathematical description of the research object, i.e., the lead–acid battery, obtained as a result of the tests, is presented below. For this purpose, Equation (9) was decoded in order to determine the coefficients describing the quantitative influence of the individual physical variables on the value of the loaded battery voltage. The decoded linear model of the research object is a quantitative model that describes the dependence of the loaded battery terminal voltage U, as the dependent variable, on the nominal (rated) capacity Q, the discharge current I, the ambient (electrolyte) temperature T, and the (technical) state of charge (SoC) k, as the independent variables. This is presented as Equation (10). This makes it possible to know the "degree of influence" of the individual independent variables on the dependent variable.

$$\mathbf{U} = 10.5\mathbf{3} + 0.0036\mathbf{Q} - 0.0052\mathbf{I} + 0.012\mathbf{T} + 1.033\mathbf{k} \,\mathrm{[V]}.\tag{10}$$

Thus, using the principles of experiment planning, a multidimensional model of the acid battery under the stationary operating conditions was obtained. This method significantly reduces the time needed to conduct experiments in order to achieve the intended research objective, especially when the objective is to develop a mathematical model with a known (assumed) form that describes the relationships between the factors.

#### *3.2. Acid Battery Non-Stationary Operating Characteristics*

In the electrical circuit of the acid battery under the non-stationary operating conditions, the dynamic characteristics of the battery are revealed, which can be represented by the variability of its internal resistance. On the basis of Equation (1), the internal resistance of the analyzed battery was determined (Figure 5), considering that the voltage at the terminals of the unloaded battery, i.e., its electromotive force, was equal to 12.97 V.

**Figure 5.** The changes in the internal resistance of the loaded battery.

The analytical form of the obtained dependencies (internal resistance and analogously of the voltage and current consumption) is convenient for the engineering calculations for predicting the features of an object. One of the significant problems in this case is the choice of the form of a regression function that is appropriate as an object or a process model. In the analyzed case, e.g., for the internal resistance, it is advisable to adopt the exponential function featuring the form Equation (11) because of the nature of the variability of the observed dependency. In addition:


$$\mathbf{R}\_{\rm int} = \mathbf{R}\_{\rm s} - \mathbf{R}\_{\rm V} \exp\left(-\frac{\mathbf{t}}{\tau}\right),\tag{11}$$

where:


The obtained signal courses (Figures 1–3, and 5) indicate the need to isolate fixed and variable parts of the dependences. Clear determining the value of the specified course is difficult because, under exponential variability, this value is reached in infinity. In addition, especially at high current values, low temperature, and poor battery condition, the changes in the value of the analyzed signals can also be a result of the battery discharge, and thus a permanent change in its properties.

The variable part of the course, as presented in Figure 5, cannot be easily described using one exponential function. In this case, the description can be made using a glued function, i.e., a set of exponential functions defined in the different time intervals. The functions should meet the condition of continuity at the limits of the time intervals. The general form of the glued function, F, and the continuity condition can be written as in Equation (12):

$$\begin{array}{c} \mathbf{F(t)} = \mathbf{F\_i(t)}; \ t\_{i-1} \le t < t\_{\mathbf{i}}\\ \mathbf{F\_i(t\_i)} = \mathbf{F\_{i+1}(t\_i)}; \ \mathbf{i} = 1, \ldots, \mathbf{n} - 1. \end{array} \tag{12}$$

In this case, another problem is the choice of the number and domain of each function, which are related to the description complexity and accuracy. As a criterion for the choice and assessment of these properties, the coefficient of determination R<sup>2</sup> for an individual function can be used. With regard to the analyzed dependencies in Figure 6, the fixed voltage values at the loaded and unloaded battery terminals were determined. Using the value of the coefficient of determination as a criterion, the time intervals of voltage

stabilization at the terminals of the loaded and unloaded battery were determined, in addition to the variation intervals, for which the voltage change characteristic has an invariable value with respect to the time constant (Figure 6). In this manner, it was established that the voltage stabilization time of the loaded battery is approximately 4.5 s, and the stabilization of the voltage value after the load was turned off did not end within 20 s. It is important to note that the approximation of the value of the variable voltage of the loaded battery using the spline function requires two components, and for an unloaded battery, the required number of components is equal to three.

**Figure 6.** The variable component of the: (**a**) loaded battery voltage together with the approximating glued function lines (regression equations are written in the text (13)); (**b**) unloaded battery voltage together with the approximating glued function lines (regression equations are written in the text (14)).

The final form of the analytical description of the two dependencies is presented by Equations (13) and (14), and its illustration, together with the exponential analytic functions, is shown in Figure 6.

$$\mathbf{U}\_{\text{load}}(\mathbf{t}) = \begin{cases} 0.68 \exp(-3.47 \mathbf{t}) \ 0 \le \mathbf{t} < 0.12; \\\ 0.55 \exp(1.17 \mathbf{t}) \ 0.12 \le \mathbf{t} < 4.5. \end{cases} \tag{13}$$

$$\mathbf{U}\_{\text{unload}}(\mathbf{t}) = \begin{cases} 0.65 \exp(-2.01 \mathbf{t}) \, 0 \le \mathbf{t} < 0.25; \\\ 0.41 \exp(-0.28 \mathbf{t}) \, 0.25 \le \mathbf{t} < 1.7; \\\ 0.32 \exp(-0.14 \mathbf{t}) \, 1.7 \le \mathbf{t} < 20. \end{cases} \tag{14}$$

The second possible means of describing the presented dependencies with the exponential functions is using their linear combination, i.e., a mixture of exponential functions. The mixture of functions, Fi, can be undertaken as follows Equation (15):

$$\mathbf{F(t) = \sum\_{i=1}^{n} a\_i F\_i(t)}\tag{15}$$

where ai represents the function weighting factors, which are also amplitudes of each individual function.

In the case of the analyzed battery, the description was made using a mixture of the voltage characteristic curve functions within the time interval from 0 to 4.5 s for a loaded battery. A stable component of the value of *U*<sup>s</sup> = 11.37 was extracted. Hence, a very good correspondence of the description with the real dependency was obtained, by distinguishing the range of fast polarization voltage variations in the period up to 0.1 s. In this case, a description according to Equation (16) was obtained, and the separated intervals and their approximation functions are shown in Figure 7a. Similarly, the voltage dependencies at the battery terminals after switching off the load were described using a mixture of exponential functions (Figure 7b, Equation (17)). In addition, in the case of

a mixture of functions, it was necessary to use the three components of the exponential function.

$$\mathbf{U}\_{\rm load}(\mathbf{t}) = 0.19 \exp(-30.88 \mathbf{t}) + 0.55 \exp(-1.17 \mathbf{t}). \tag{16}$$

$$\mathbf{U\_{unload}(t)} = 0.2548 \exp(-26.12 \mathbf{t}) + 0.21 \exp(-2.12 \mathbf{t}) + 0.32 \exp(-0.14 \mathbf{t}). \tag{17}$$

**Figure 7.** Determined voltage components at the terminals of the: (**a**) loaded battery within the time range up to 4.5 s (regression equations, mixture of exponential functions, are written in the text (16)); (**b**) unloaded battery within the time range up to 20 s (regression equations, mixture of exponential functions, are written in the text (17)).

Attention should be paid to the significant, more than 25-fold, differentiation in the time constants of both functions (for the loaded battery), which is equal to about 0.032 s for the fast-changing component, i.e., in the time interval up to 0.1 s and 0.86 s for the slow-changing component (they are equal to the inverse of coefficients specified in the function exponents).

#### *3.3. Battery Structural Model*

In the previous considerations, according to the electrical diagram given in Figure 4, the reason for the voltage change at the terminals of the loaded battery was recognized to be the change in its internal resistance. The primary reasons for the change are the changes in the electrolyte density around the electrodes and in the inner layers of the active mass of the battery plates. The change in the electrolyte density is also the reason for the changing potentials of the electrodes, i.e., the electromotive force of the battery. Therefore, the change in the voltage at the terminals can also be considered to be the change in the component of the electromotive force called the electromotive force of polarization. The polarity of each electrode (anode and cathode) can be distinguished, depending on the location of the polarization processes and the voltage drop in the electrolyte. In general, the electrical resistance of the battery is constituted by resistance, capacitance, and inductance components.

The electrical diagram of the acid battery shown in Figure 4 can be used to describe the operation of the battery under a constant current load or constant resistance. The variability of the internal resistance with time under dynamic load conditions makes it practically impossible to use this diagram to determine the response of the accumulator to the variable, dynamic force.

The description of the battery discharge characteristics (voltage at its terminals) using the exponential function enables the introduction of the electric components to the equivalent battery circuit, whose electrical properties generate responses in the form of exponential function. Such a component of vicarious battery diagrams may consist of a capacitor and a resistor through which the capacitor is charged or discharged. Therefore, it is possible to connect the RC circuits to a stationary source of the electromotive force of the battery EB, as in Figure 8. Under variable load conditions, these circuits generate the electromotive force of the polarization components Epi (Figure 8) according to Equations (16) and (17). A description of the battery discharge characteristics by means of a linear combination of two or more exponential functions indicates the possibility and need to also apply a larger number of RC circuits connected in series to the equivalent circuit of the battery.

**Figure 8.** Electrical equivalent circuit for an acid battery.

Determination of the parameters of the circuit components is made possible on the basis of the test results of the battery discharge characteristics, as specified in Equations (16) and (17). It is known that the capacitor discharge characteristic is the exponential curve in the following form Equation (18):

$$\mathbf{U}(\mathbf{t}) = \mathbf{U}\_0 \exp(-\frac{\mathbf{t}}{\pi}) = \mathbf{U}\_0 \exp(-\frac{\mathbf{t}}{\mathbf{RC}}).\tag{18}$$

This enables physical values to be assigned to the indicated components of the battery equivalent circuit. This is due to the fact that the time constant of the discharge process is τ = RC.

#### **4. Discussion**

The characteristic of a device functioning (operating) is the dependence of its output signal feature value, i.e., device response, on the value of the input signal. Devices have static and dynamic characteristics. A device's static characteristic is most often the function y = y(u), which represents the dependence of the value of the output signal feature y of the device on the input value, i.e., the input signal feature u under the steady-state conditions. An example of such a characteristic may be the dependence of the voltage at the battery terminals on the factors influencing its value, as presented above. The most frequently expected static characteristic is a linear one-dimensional or multi-dimensional characteristic, for example, as shown in Equation (10).

The dynamic characteristic of the device determines the transformation of the input signal u(t) (extortion signal), which varies as a function of time, into the output signal y(t); that is, the variable as a function of time constitutes a response of the system to this input. The dynamic characteristics of a device are most frequently described using transmittance. The operator transmittance, also referred to as a function of the device transition, is the ratio of the Laplace transform of the output signal Y(s) to the Laplace transform of the input signal U(s) under the zero initial conditions Equation (19):

$$\mathbf{G(s)} = \frac{\mathbf{Y(s)}}{\mathbf{U(s)'}} \tag{19}$$

where the Laplace transformation is a transformation of the time function f(t) into a complex function of the complex variable F(s) Equation (20):

$$\mathbf{F(s)} = \mathcal{L}\{\mathbf{f(t)}\} = \int\_0^\infty \mathbf{f(t)e^{-st}} \mathbf{d}t. \tag{20}$$

On the basis of the transition function, using the inverse Laplace transform, it is possible to determine the signal that will be obtained at the output of the system for any input signal Equation (21)—or vice versa—to determine the driving signal, which should be given at the input of the device to obtain the desired response of the device.

$$\mathbf{y}(\mathbf{t}) = \mathbf{L}^{-1}\{\mathbf{U}(\mathbf{s})\mathbf{G}(\mathbf{s})\}. \tag{21}$$

The transmittance is most often presented in the form of the amplitude characteristic |A(f)| and phase characteristic Φ(f) of the device as a function of frequency f or angular frequency ω, i.e., amplitude and phase transfer in the event of a sinusoidal input. A lack of linearity of the amplitude and phase characteristics may cause dynamic deviations of the output signal. The amplitude and phase characteristics of the device are derived from the spectral transmittance.

The spectral transmittance is the ratio of the value of the complex Y response of the system caused by a sinusoidal extortion to the complex value of this extortion in the stationary state. Thus, spectral transmittance characterizes the response of the system to a sinusoidal extortion. The spectral transmittance can be obtained from the operator transmittance by substitutings=jω, thus obtaining the corresponding Fourier transform.

According to the aforementioned dependencies Equations (19) and (20), the battery operator transmittance was determined for the signal presented in Figure 7a and described using Equation (16). It was assumed that the input signal is the leap in the polarization electromotive force with a value equal to the sum of component amplitudes in Equation (16), given as Equation (22):

$$\mathbf{E\_{p0}} = \mathbf{E\_{p10}} + \mathbf{E\_{p20}}.\tag{22}$$

Therefore, in accordance with the above, the output signal Equation (16) can be written as Equation (23):

$$\mathbf{E\_{P}} = \mathbf{E\_{P}} \mathbf{1}\_{0} \left[ 1 - \exp\left(-\frac{\mathbf{t}}{\tau\_{1}}\right) \right] + \mathbf{E\_{P20}} \left[ 1 - \exp\left(-\frac{\mathbf{t}}{\tau\_{2}}\right) \right]. \tag{23}$$

Performing the Laplace transforms of the above-written expressions, the transformations of the input and output signal were obtained in the forms of Equations (24) and (25):

$$\mathbf{E\_{p0}(s)} = \frac{\mathbf{E\_{p10}} + \mathbf{E\_{p20}}}{s}.\tag{24}$$

$$\mathbf{E\_{P}(s)} = \frac{\mathbf{E\_{p10}}}{\mathbf{s}(1+s\mathbf{r}\_{1})} + \frac{\mathbf{E\_{p20}}}{\mathbf{s}(1+s\mathbf{r}\_{2})}.\tag{25}$$

As stated above, the battery operator transmittance can be obtained by dividing the Laplace transforms of the output signal and the input signal. Thus, after performing the appropriate transformations, the transmittance of the battery can be presented in the form Equation (26) as below:

$$\mathbf{F(s)} = \frac{1}{\mathbf{E\_{p10}} + \mathbf{E\_{P20}}} \cdot \frac{\mathbf{E\_{p10}}(1 + s\mathbf{\tau\_2}) + \mathbf{E\_{p20}}(1 + s\mathbf{\tau\_1})}{(1 + s\mathbf{\tau\_1})(1 + s\mathbf{\tau\_2})}. \tag{26}$$

Knowledge of the operating characteristics of the acid battery, both static and dynamic, in the form of its transmittance Equation (26), enables determination of the battery response, i.e., the voltage at its terminals, for any load value, i.e., the current drawn from the battery. In the expression describing the static characteristic Equation (10), the independent factors include the technical condition, i.e., the state of charge of the battery, k (it is advisable and necessary to determine similar relationships for the dynamic characteristics). The comparison of the battery voltage value determined on the basis of the characteristic for the reference battery, with the voltage value measured under the experimental conditions, may enable the (SoC) value k of the researched battery to be determined. In this case, value k becomes i.e., the battery status indicator, and can be used as a diagnostic parameter for the determination of the status of the battery. It is expected that the developed dependencies

will be used in the diagnostic procedure of the battery, based on the measurement of the engine start signals, i.e., the driving of its crankshaft by the starting system. The proposed method will allow not only the diagnosis of the condition of the acid battery, but also the electric starter and the internal combustion engine in terms of the resistance to motion, taking into account the compression pressure of cylinder charges.

#### **5. Conclusions**

The acid battery is a functionally and structurally complex non-linear power source, whose features are dependent on many parameters. Its static characteristic, i.e., the loaded battery terminal voltage U in stationary operating conditions, depends on the nominal (rated) capacity Q, the discharge current I, the ambient (electrolyte) temperature T, and the (technical) state of charge (SoC) k as the independent variables. For the purpose of describing the dynamic characteristic of its operation, that is, the response to a rapid leap in current (i.e., a load with a constant current or resistance), it is convenient to use the exponential functions in the form of glued functions or a mixture of functions. Both methods of description correspond to two different electric models (equivalent circuits) of the acid battery in the form: (1) the electromotive force and the variable internal resistanc; (2) the stationary electromotive force and the (two or three) RC systems having different characteristics, resulting in changes in the electromotive force of the polarization of the battery in the circuit. The presence of two or three different components of the electromotive force of polarization indicates that the equivalent circuit of a lead–acid battery should include at least two RC circuits connected in series, with the significantly different parameters defined by means of time constants. In fact, the change in the time constant of the polarization electromotive force occurs continuously, from very small values to theoretically equal to infinity. The consideration of many independent parameters in the description of the battery and its structure requires long-term extensive experimental research.

A feature of modern machine exploitation is the constant, systematic increase in the role and meaning of technical diagnostics. The broad possibilities of its application result from the change in the properties of the exploitation objects, including motor vehicles and the development of methods and means of diagnosis using digital signal recording and processing techniques. In the diagnostics of internal combustion engines and their starting systems, the diagnostic parameters of the working and accompanying processes of the driving of the crankshaft can be used. These diagnostic parameters include: the current consumed by the starter, the voltage at the battery or starter terminals, and the speed of the crankshaft forced by the starter. The set of electric starter characteristics depends on the properties of the energy source, i.e., the acid battery.

The developed models, both for the stationary and non-stationary conditions, will be used in the proposed and currently developed diagnosis method for the internal combustion engine–electric starting system based on the engine start-up signals and its driving by the starting system. It will be possible to determine the state of charge of the battery on the basis of Equation (10). The dynamic components of the model in the form of transmittance Equation (26) will allow determination of the significance of the influence of the battery's dynamic characteristics on the process of driving the crankshaft of the engine, and the necessity to include this in the diagnostic test of the system.

As a result of the presented research and characteristics of the acid battery, and the analyses performed, conclusions important for the knowledge and understanding of the principles of operation of the acid battery can be formulated:


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

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Exclude this statement.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

