Controlling the Unbalanced Voltages of a Series-Connected Lead-Acid Batteries in a PV Power Storage System using Dynamic Capacitor Technique

L ead-acid batteries have been used increasingly in recent years in solar power systems, especially in homes and small businesses, due to their cheapness and advanced development in manufacturing them. However, these batteries have low voltages and low capacities, to increase voltage and capacities, they need to be connected in series and parallel. Whether they are connected in series or parallel, their voltages and capacities must be equal otherwise the quality of service will be degraded. The fact that these different voltages are inherent in their manufacturing, but these unbalanced voltages can be controlled. Using a switched capacitor is a method that was used in many methods for balancing voltages, but their responses are slow. To increase the response and control of the balancing process, this research proposes a novel technique that consists of a dynamic capacitor for controlling the unbalanced voltages of series-connected lead-acid batteries. The proposed technique uses a main capacitor and an inductor with two switches their on/off states are controlled through a pulse width modulation. The technique is designed and validated using MATLAB/Simulink and the results for different cases are compared with other techniques such as switched capacitor technique. Results show that the proposed method promised the balancing control in a shorter time and better performance than other techniques which are crucial in the battery’s voltage balancing.


INTRODUCTION
Recently, the demand for photovoltaic (PV) technology in producing electricity has increased substantially (Victoria et al., 2021;Zhang et al., 2021;IEA, 2022).On the other hand, the continuously reducing costs for building PV power plants make them successfully be used to supply electricity not just for large areas but for small businesses and households as well.However, the intermittency behavior of PV power systems especially during the night and cloudy days remains the main drawback of them.A battery storage system is counted as one of the best practical solutions to the PV drawback (Salameh et al., 2022; Apribowo, 2022).In the last decade, the growth of PV power systems and the development of electric vehicles have made huge progress in adopting new technologies for manufacturing different types of batteries (Chen et al., 2020).Batteries now have more charge/discharge cycles, longer life, stronger, higher capacities, and low costs (Lipu et al., 2022;Pan et al., 2022).However, these batteries are still available in low voltage mostly 12 DC voltage that need to use more batteries in series and parallel connection combination to increase their voltages and capacities to follow the ratings of inverters in PV power systems (Manimekalai et al., 2013;Fortenbacher et al., 2017;Bagalini et al., 2019, Wu et al., 2022).One of the most drawbacks of almost all types of batteries is self-discharging.This selfdischarging reduces battery voltage and its impact will be worse when batteries are connected in series and parallel.This impacts the operation of inverters, decreases the storage capacity, decreases cycle charging/discharging, and shortens the battery's life.In general, parallel operation of batteries may balance their voltages to the level that all have the same voltage.While in series connecting, self-balancing does not achieve, therefore, an external procedure is required to maintain the same voltage of all batteries (Wang, et al.

PROPOSED BALANCED BATTERY VOLTAGE SYSTEM
The proposed voltage balancing system for lead-acid batteries connected in series is depicted in Fig. 1.The system consists of lead-acid batteries connected in series, these batteries have the same capacities, rated voltages, same type, and same manufacturer.A voltage detector is used to detect the voltages of all batteries for any inequality.The pulse Width Modulation (PWM) controller sends signals to the switches in the dynamic capacitor according to the rate of the imbalanced voltage detection (Alvarez et al., 2020).The dynamic capacitor circuit consists of the capacitor, inductor, and switches to process the balancing in the batteries' voltages.
where Ke is the voltage temperature coefficient of the battery, The is the electrolyte temperature in which (273+The) is temperature measured in Kelvin, and SOC is the state-ofcharge which is given by: where Qe is the battery consumed charge in Ah, and C10 is the rated charge capacity of the battery in Ah .
The cell main branch resistors Ri1 and Ri2 are given by: where R10, R20, A21, and A22 are battery internal parameters and they are constant for a particular battery, Im is the actual main branch current, Io is the rated battery cell current, and DOC is the depth of charge and is given by (Collath et al., 2022); where C1 is the actual battery capacity under the actual discharge current and is given by: where τc is the time constant of the cell.
The output resistance of the equivalent circuit, Ro, is given by (Maraud et al., 2016): where Roo and Ao are constant parameters .
The parasitic branch current, Ip, is given by (Maraud et al., 2016); where Epo, Gpo, and Ap are battery cell constants, Ep is parasitic branch voltage, and Thf is the electrolyte freezing temperature (-40 O C).

DYNAMIC CAPACITOR
The main components of the dynamic capacitor are an inductor, L, a capacitor, C, LinCin filter, and two switches, S1 and S2.The electrical circuit diagram for the dynamic capacitor is shown in Fig. 3 (Aula, 2022).The capacitor voltage, vc, can be computed from the following relationship (Aula, 2022); where   is the input voltage (supply voltage), and D is the duty cycle and is given by; where , Rf is a ripple factor of the inductor current, xc is the capacitive reactance of C, and f is the switching frequency, which is supplied by pulse width modulation (PWM).

CONTROLLING UNBALANCED BATTERY VOLTAGES
The principle of controlling the unbalanced voltages of batteries, in this research, is using a dynamic capacitor by dividing the LC filter according to the number of batteries in series.A proposed technique for controlling the unbalanced voltages of two series-connected batteries using a dynamic capacitor is depicted in batteries' unbalanced voltages.

Mode1
The first mode of controlling the unbalanced voltage represents switch S1 ON and switch S2 OFF, the simplification circuit diagram of mode 1 is shown in Fig. 6.The current passed through the inductor, no energy storage in this mode, from both batteries can be represented as: where vb1 and vb2 are battery voltages, iL and vL are inductor current and voltage in mode 1, and R1 is the inductor internal resistance.

Mode 2
In the second mode, switch S1 is OFF and switch S2 is ON, and the equivalent circuit diagram that represents this mode is shown in Fig. 7.The stored energy in the inductor in mode 1 in time t will increase the capacitor voltage vL, which can be expressed by the following statespace representation: where vc and iL are capacitor voltage and inductor current in mode 2. In which iL = -iC (capacitor current), and R2 is the summation of inductor and capacitor internal resistance.
The eigenvalues of the system in mode 2 are given by: From Eq. ( 16), the undamped natural frequency is 1/√LC rad/sec, and the damping ratio, ζ, . Usually, the capacitor value is much smaller than the inductor, thus, the two eigenvalues become: Hence, the response of the second order system and according to the damping ratio is the underdamped response which undamped natural frequency decays exponentially.The inductor current in mode 2 after solving the system Eq.( 15) with the assumption that capacitor voltage is initially zero can be represented by the following expression: where where iLu represents the unforced inductor current, iLf represents the forced current .
The capacitor voltage is given by: where where vcu represents the unforced capacitor voltage, vcf is the forced capacitor voltage, and D is the duty cycle for switching ON/OFF as defined in Eq. ( 10).

L V b1
V b2 diagram.

Modes 3 and 4
After the second mode in which the second cycles of switches begin, the inductor current, iL, in Eq. ( 18) becomes the initial current to Eq. ( 13), and the same occurs for inductor voltage, vL.Also, the capacitor voltage in Eq. ( 21) will be the initial capacitor voltage in the consecutive cycles.The sequence of charging/discharging capacitor voltage according to both switches, S1 and S2, at 5kHz PWM pulses are shown in Fig. 8.The switching ON/OFF operations which control the inductor and capacitor voltages proceed through pulse width modulation (pulses).The rate of these pulses (pulses per second) determines the inductor voltage and capacitor voltage simultaneously.

SIMULATION AND RESULTS
For most solar power systems, lead-acid batteries for their cost affordable, and acceptable capacities effectivity are used as backup storage.Therefore, in this research, lead-acid batteries were used with a capacity that matches the currently commercially available ones.A 12 V, 200Ah is used for all batteries throughout this research, and their parameters are given in Table 1.Fig. 9 shows the MATLAB/Simulink model which is implemented for validating the model and method proposed in this research.For the optimum controlling of both switches, the PWM is chosen to be 5kHz.The inductor and capacitor values, L and C, are 3mH and 200μF.The internal resistances for both the inductor and capacitor are set to 0.005Ω and 0.003 Ω, respectively.The filter inductor and capacitor are set to 0.03mH and 35μF, respectively.It is wealth to note that, for filtering only one inductor is sufficient to be used, and the number of capacitors depends on the number of batteries in the series, for example, for two batteries, two filleter capacitors are used, and for four batteries in series, four filter capacitors are used, and so on.Fig. 4 is an example of how these elements were connected to batteries .It is wealth to note that for either technique, voltages of lead-acid batteries can be balanced in a shorter time when the PWM pulses are increased.However, this is left for future work to be studied and implemented.

CONCLUSIONS
In this study, a dynamic capacitor is proposed for controlling the unbalanced voltages of lead-acid batteries that are mostly used in residence as well as small business solar power systems to back up the surplus power and to be used later mainly during the night.The details of the mathematical model of the lead-acid batteries and the operation of the dynamic capacitor technique were presented.The dynamic capacitor technique consisted of two switches in which one is on while the other is off and vice versa during each cycle of operation.The PWM is used for controlling the ON/OFF of both switches.The simulation was carried out using MATLAB/Simulink to validate the proposed technique.Simulation results showed the effectiveness of the proposed technique in which the voltages of all lead-acid series-connected batteries were balanced in a shorter time as other techniques can provide such as the switched-capacitor technique.The balancing voltages in two batteries reached the settling time within 100 seconds for as low as 4% differences, while these took above 500 seconds and just a 20% difference in the switched-capacitor method.

NOMENCLATURE
Different techniques have been used for equalizing batteries' voltages.(Kim, et al., 2014) presented a switched capacitor and the method was applied to lithium-ion batteries.(Ye, et al., 2017) proposed a series of switched-capacitors for battery and super-capacitor balancing strings and to increase the speed of balancing different topologies were proposed with resistors.A bi-directional Cuk converter was presented by (Zheng et al., 2018; Rasheed, 2020), and a fuzzy control method was used for controlling the Cuk converter for voltage balancing of lithium-ion batteries.(Moghaddam and Van den Bossche, 2019; Ho, et al., 2021; Wang et al., 2023) presented different types of switched capacitor models in which these models were simulated on lithium-ion batteries operating at voltages 3 to 4.2V (Zau et al., 2022).However, many studies have been done on unbalancing lithium-ion batteries but a few are dealing with lead-acid batteries.Charge equalization systems were proposed for serial lead-acid batteries in hybrid power systems (Belmokhtar et al., 2016; Akash and Sumana, 2021), two systems were included: active and passive systems, the passive was based on a resistor element to remove the excess charge until matching high voltage batteries to lower voltages.Recently, lead-acid batteries have been widely used in households' electricity storage systems due to their lower prices, and availability in different capacities, moreover, new technology in manufacturing these types of batteries makes them last longer (Dufo-López, et al., 2021; Rajanna and Kumar, 2021).With the growth of using lead-acid batteries, especially in PV storage systems, and they are subjected to voltage differences during operations, therefore, an efficient voltage balancing technique is highly recommended for retaining the voltage balance of all batteries in a shorter time in low power consumption (Geoffrey et al., 2018; Kavaliauskas et al., 2023).This study proposes a dynamic capacitor technique for controlling the unbalanced voltages of lead-acid batteries which are connected in series.The proposed technique is applied on a 12V, 200Ah lead-acid battery and simulated on MATLAB/Simulink® for validating the results, and results are compared to the switched-capacitor technique.The result outcomes for different cases show that the proposed technique achieves the voltage balance in a shorter time than the other technique.The technique can be implemented on most batteries in PV power storage systems in residences and small businesses.

Figure 1 .
Figure 1.Block diagram of the balanced voltage system
ton and toff are switching times for both switches accounted when S1 is on, S2 is off, and vice versa.The values of C and L are computed by (Aula, 2022);

Figure 8 .
Figure 8. Capacitor and inductor voltage correspond to switch operations.

Figure 9 .
Figure 9. MATLAB simulation model for two lead-acid batteries in series with dynamic capacitor implementation.

Figure 10 .
Figure 10.Balancing of two batteries' voltages connected in series for SOC 90% case.

Figure 12 .
Figure 12.Balancing of two batteries' voltages connected in series for SOC case 95%.

Figure 13 .
Figure 13.Batteries SOC case 95%.In the third case, since other homes use inverters that are rated 48 V for batteries, therefore, four lead-acid batteries are used in series.Their SOCs are set as 100%, 97%, 95%, and 90%, respectively.The simulation results are shown in Figs.14 and 15.

Figure 14 .
Figure 14.Balancing of four batteries' voltages connected in series.

Figure 15 .
Figure 15.SOC for all four batteries during the balancing control procedure.

Figure 16 .
Figure 16.Two series connected batteries for SOC 90% for switched-capacitor technique.

Figure 17 .
Figure 17.Two series connected batteries for SOC 95% for switched-capacitor technique.

Figure 18 .
Figure 18.Four series connected batteries for switched-capacitor technique.