Battery State of Charge Modeling for Solar PV Array using Polynomial Regression
BB ATTERY S TATE OF C HARGE M ODELING FOR S OLAR
PVA
RRAY USING P OLYNOMIAL R EGRESSION
Siddhi Vinayak Pandey
Dept. of Electrical EngineeringAdani Institute of Infrastructure EngineeringAhmedabad, India [email protected]
Jeet Patel
Dept. of Electrical EngineeringAdani Institute of Infrastructure EngineeringAhmedabad, India [email protected]
Harsh S. Dhiman
Dept. of Electrical EngineeringAdani Institute of Infrastructure EngineeringAhmedabad, India [email protected]
August 21, 2020 A BSTRACT
Due to the increased demand of Solar PV arrays, its integration with a battery and modelling ofprecise State of Charge (SoC) is a consequential parameter to understand the available battery capacityin real time domain. In this paper, the integration of Solar PV Array with the first-order RC circuithas been implemented utilizing the MATLAB (Simulink Library). For experimentation, the OpenCircuit Voltage ( V oc ) and the Short Circuit Current ( I sc ) of the solar panel is considered as 36.3 voltand 7.84 ampere. The continuous fluctuating irradiance from 110–580 W/m leads in the variationof output voltage of the Solar PV arrays. Also, the variations of battery charging current, voltageacross battery and battery SOC due to variations in irradiance are examined in detail. The proposedmethodology of this manuscript explains the authentic time modelling of SoC utilizing the second,third, fourth, and fifth order of a polynomial regression technique. The comparative relationshipbetween the OCV and SoC during the charging of the model with Solar PV Array has been alsoobtained for different orders of polynomials. Quantitatively, 5th order model outperforms 2nd, 3rdand 4th by 27.5%, 26.7% and 21.3% respectively. Sustainable energy sources lead to reduction in carbon footprint and thus increase the reliability of a system [1, 2, 3, 4].The State of Charge is a parameter by which users can get details about the availability of battery capacity. The precisecalculation and estimation of the available energy within a battery has always been a challenging task. Mathematically,SoC is a ratio of current capacity Q ( t ) to nominal capacity Q n given as SoC ( t ) = Q ( t ) Q n (1)The precise calculation and modeling of the SoC will not only increases the system performance but also impacts cyclelife in a longer run. Measurement and estimation of SoC remains one of the challenging tasks. SoC estimation is basedon various sensor-based, filter-based, and data-driven techniques out of which Coulomb counting being the fundamentalone. In [5], authors use Coulomb counting where current is integrated with respect to time. However, this method isinefficient due to the effect of temperature, discharge current, and cycle life of the battery. In order to overcome this,filter-based, and data-driven estimation algorithms are utilized to estimate the SoC of the battery. In [6], authors use a a r X i v : . [ ee ss . S Y ] A ug PREPRINT - A
UGUST
21, 2020Kalman Filter technique for SoC estimation. The Kalman filter is an estimator used to estimate the linear and nonlinearsystems. Based on the present data of voltage, current, and temperature, it estimates the SoC with good accuracy andprecision. While the estimation error of fully and partially charged battery is found to be around 0.5%. Apart from theKalman filter, machine learning based algorithms are also used to estimate the SoC of a battery. In [7], authors usea Support Vector Machine (SVM) approach, which is a statistical learning method used for estimating the SoC of abattery. The SoC of a Lithium Iron Manganese Phosphate LiFeMnPO battery is estimated under the constant chargingand variable load condition. The error in the proposed model is less than 4%, while the RMSE is 0.4% throughout thewhole experiment. Apart from SVM, various other data driven methods such as Neural networks (NNs), Deep neuralnetworks (DNNs) and Reinforcement Learning (RL) approach are used for the estimation of SoC.Due to green gas emissions, the world is perpetually shifting towards renewable energy sources. Solar and wind energybeing the pioneers in power portfolio. The on-field efficiency of solar panels is near about 18-21% [8]. However, due tothe recent advancement in perovskite solar cells, reflectors, photonic crystals, and recycling PV cells, the efficiencyof the solar cells are increasing drastically. Further, it is expected that till 2030 the on-field efficiency of Solar basedphotovoltaic cells will be near about 30% [9].Nowadays, the integration of Solar PV arrays with batteries is gaining tremendous importance due to intermittencyposed. Regarding this, many electric conveyances, grids, space satellites are directly charging their batteries throughsolar PV arrays [10]. Mainly, two topologies exist for solar PV array with battery. The first one being stand alonesystem, while the other being the grid-tied system. In the standalone system, solar PV is connected to the load via abattery. While in a grid-tied system, the load is supplied by a grid in case of failure in PV side [11].In order to implement Solar PV arrays in applications such as electric vehicles, space satellites, and drones, the weightof the module plays a very crucial role. The huge weight of solar panels can increase the overall payload capacity ofthe vehicles and to overcome this, installation of thin-film crystalline based solar panels is suggested. The thin-filmcrystalline based solar cells are very light in weight. However, they are less efficient in their photovoltaic operation.Majority of the research is focused to enhance the efficiency of thin-film crystalline solar panels [12]. The selectionof battery while implementing the Solar PV arrays is also a consequential parameter for the efficient and vigorousperformance of the system. In [13], authors discuss the case of LFP (Lithium Ion Phosphate) and LCO (Lithium CobaltOxide) batteries which when tested at 45 ◦ C observed an optimistic operation for the photovoltaic battery integratedmodule.The energy generated through solar PV arrays is directly dependent on the irradiance and surface temperature of solarpanels as variation in irradiance and surface temperature can impact power generation of the solar PV plant [14]. Thistype of change can directly affect the voltage and current level of the plant. Furthermore, this also impacts the SoClevel of the interconnected battery. In [15], authors proposed a study based on the effect of variation in irradianceand surface temperature for solar PV arrays. In the proposed model, factors such as fill factor, efficiency, open-circuitvoltage, and maximum power are the parameters get affected due to the variation in irradiance and surface temperatureof a solar predicated PV arrays. The increment in irradiance increases the open-circuit voltage and overall efficiencyof the system. While increase in the surface temperature of the PV cell decreases the open-circuit voltage, fill factor,efficiency, and a maximum power of a system. Apart from these factors, the construction of a battery is dependent uponthe materials, modeling, and effect of the ageing mechanism during the real-time charging and discharging of a battery.Due to the escalating demand for energy storage systems, batteries play an important role in storing bulk power. As amatter of fact, an elevated research is going on in the field of eco-friendly based battery material that gets recycledeasily. variants such as Lead acid, Nickel, Lithium, Cadmium, alkaline, Mercury, and Nickel Metal Hydride are toxicelements often used in fabrication process. A summary of the methods for estimating the SoC of a battery, dynamicbattery modeling, and ageing mechanism of Li-ion battery is depicted in Table 1.2 P R E P R I N T - A UGU S T , Table 1: Brief summary of current research in battery energy storage systems
SoC estimation [16] Dynamic battery modeling [17] Ageing mechanism [18] • Direct measurement methods • Electrochemical model • Loss of active material1. Coulomb counting It uses electrochemical parameters The positive electrode of the battery undergoes contraction2. Open-circuit voltage and variables to define a battery model and relaxation constantly which causes the material to deteriorateslowly loses lithium absorbing capacity.• Filter-based methods • Equivalent first-order battery model • Loss of Lithium inventory1. Kalman filter The first-order model has a resistance R While forming the SEI layer large amount of lithium is consumed which2. Extended Kalman filter which represents internal battery resistance is the main reason for LLI. The high rate charging and dischargingof the battery along with one RC network connected with it or low-temperature, lithium plating may occur on thenegative electrode and form lithium crystal brancheswhich cause further loss of Li-ion.• Data driven methods • Equivalent second-order battery model1. Support vector machines A Second-order system has a similar circuit Almost all positive electrodes are made up of carbon.2. Neural Network connected with it. An SEI(solid electrolyte interface ) layer is formedbut with an extra (one more) RC network and diminished during the charging and discharging process of the battery. PREPRINT - A
UGUST
21, 2020In this manuscript, the first-order RC circuit is implemented under the variation of irradiance on the Solar PV array. Thevariation of SoC during the charging of a battery has been obtained using the Kalman Filter technique, and based on thecharging response of a first-order model, the second, third, fourth and fifth degree of polynomial regression technique isused to obtain the relationship between OCV and SoC. The comparative analysis of regression between the OCV andSoC at different degrees of the polynomial is investigated in detail.The remainder of the manuscript is organized as follows. Section 2 describes the simulation setup of our proposedmethodology. Section 3 gives insights about modeling and comparisons of regression techniques. Section 4 providesthe result of our proposed methodology. While the acknowledgment, conclusions, future works, and references arepresented in Section 5.
In order to simulate the dynamic characteristics of a battery, the values of an input voltage source, resistor, and capacitorplay an important role [19]. In this manuscript, the first-order RC model has been used to store the energy generated bythe Solar PV array. Given the simulation setup, the dynamic first order model of the battery is illustrated in Fig. 1.Figure 1: First order battery modelIn Fig. 1 the terms V oc and V batt are open-circuit voltage and terminal voltage, R is the internal resistance while R and C are the external polarizing resistor and capacitance of the battery. During the operation, the capacitor getsenergized and stores the energy which was supplied through voltage source ( V s ). This dynamic model works in closeapproximation to the battery being used in real-time applications. Mathematically, charging of a capacitor with respectto time is given as V batt = V oc − IR e − t/τ (2) i ( t ) = V batt R e − t/τ (3)The terms V batt and i ( t ) in (2) and (3) represent voltage across and current through the capacitor. While the τ = R C denotes the time constant of the model. This time constant requires five steps (five times constant) to charge thecapacitor. During the charging conditions, the demeanor of the system remains exponential. The proposed methodology of our experimental setup aims to determine the relationship between the State of Charge(SoC) and Open Circuit Voltage (OCV) of a battery. The direct charging of a battery using the Solar PV arrays underthe variable irradiance and surface temperature of a photovoltaic cell is used to charge the battery. The proposedexperimental setup is explained using the following block diagram.4
PREPRINT - A
UGUST
21, 2020Figure 2: Block diagram for the simulation setupThe schematic block diagram in Fig. 2 consists of basic blocks which are interconnected together to perform theintended task. The Solar PV Array is kept under the variable irradiance and temperature. The variability betweenirradiance and temperature affects the output voltage generation of a solar PV array. This variable voltage is directlystored in the battery.The battery connected with solar PV array gets charged when a sophisticated amount of voltage generated across theSolar PV array. The measurement block is interconnected with such sensors, which can measure the value of voltage,current, and temperature of a dynamic battery model during the charging condition. These measurements are clearlyvisible on the display screen; which is connected next to the measurement block. In order to estimate the SoC of abattery during the charging condition; the SoC Estimator is used. In our case, the Kalman Filter is used for the SoCestimation of a battery during a direct charging condition with a Solar PV array.The Kalman Filter is a state estimator, used to estimate and track the data in the real-time domain. In our case, it takesthe input current of a battery during the charging condition with a Solar PV array for estimating the SoC of a battery inthe real-time domain. The Kalman filter takes the State-space values and inputs current of a battery. And based on thesedata, it estimates the SoC of a battery. The mathematical representation of the Kalman filter [20] is as x k +1 = A k x k + B k u k + w k y k = C k x k + D k u k + v k , (4)where A k , B k , C k and D k represent state matrices of the first order battery model. The term x k represents initial state(state vector) while y k is the set of observed data (observable at time k) in Kalman filter. During these estimations, it isrequired to know the noise of the initial system by which the filter received data for estimation. The term w k representsthe process noise and v k represents the measurement noise in a system. Due to modelling approximations and modelintegration errors, the process noise ( w k ) is a consequential parameter to include in a model. Initialization for, k = 0 , set ˆ x +0 = E [ x ] P +ˆ x, o = E (cid:104)(cid:0) x − x +0 (cid:1) (cid:0) x − x +0 (cid:1) T (cid:105) State estimate time update given as ˆ x − = A k − ˆ x − k − + B k − u k − (5)Error covariance time update P − ˆ x, o = A k − P + X − A Tk − + Q w (6)Kalman gain matrix k k = P − ˆ x,k C Tk (cid:2) C k P − ˆ x C Tk + R v (cid:3) T (7)State estimate measurement update Q ∗ k = 8 − k + K k (cid:2) y k − C k · ε − k − D k u k (cid:3) (8)Error covariance measurement update P +ˆ X k = ( I − K k C k ) P − ˜ X k (9)The state space matrices for the first-order battery model are given as x = v x + SoC x − SoC x = (cid:34) (cid:35) y = (cid:20) V batt x SoC (cid:21) u = (cid:20) i d i c (cid:21) PREPRINT - A
UGUST
21, 2020 A k = (cid:34) − / ( R C ) 0 00 0 00 0 0 (cid:35) B k = (cid:34) /C /C /C cap η c /C cap (cid:35) C k = (cid:20) γ γ (cid:21) D k = (cid:20) R R (cid:21) Table 2: Model Parameters for SoC Estimation [20]Parameters Values R R . C . C cap γ . SoC η c . . The entire simulation setup has is analyzed in MATLAB 2015b, core-i5 CPU environment. For simulation analysis, a“1Soltech 1STH-215-p” Solar PV module with the connection of 3 parallel and 1 series string. The maximum power ofthe selected module is 213.15 W. The I-V and P-V characteristics of the PV array is illustrated in Fig. 3.Figure 3: I-V and P-V curve for Solar PV arrayIn order to replicate the practical scenario, the irradiance level for PV array is kept variable rather than a constant one.Fig. 4 illustrates the solar irradiance (W/m ) for simulation environment considered in this analysis. The irradiance plotin Fig. 4 is shown for first six seconds of the simulation. 6 PREPRINT - A
UGUST
21, 2020 I rr ad i an c e ( W / m ) time (seconds) C u rr en t ( A ) V o l t age ( V ) time (seconds) S o C Figure 4: Irradiance, Current, Voltage and SoC for the simulation setup7
PREPRINT - A
UGUST
21, 2020It is evident that output voltage and current for solar PV array will vary due to the variation of irradiance. Thesevariations directly affect the charging characteristics of our dynamic battery model. In this section, a relationshipbetween voltage and SoC of a battery utilizing the polynomial regression approach is obtained. The curve fittingutilizing the polynomial regression technique engenders the values of the coefficient for the equations, which definesthe relationship between the two curves given as an input. Polynomials of order second, third, fourth, and fifth-degreeof polynomials are analysed based on their R and RMSE values.Table 3: Performance metrics for regression modelsDegree of the polynomials R RMSE2 0.7278 0.22893 0.7309 0.22764 0.7532 0.21795 0.8326 0.17958
PREPRINT - A
UGUST
21, 2020Figure 5: Curve fitting for OCV-SoC based on 2, 3, 4 and 5 order polynomials9
PREPRINT - A
UGUST
21, 2020The value of R for degree 2 is 0.7278 while for degree 5 it is 0.8326. Further, RMSE of 0.2289 for degree 2 and 0.1795for degree 5 is observed. The term R gives an idea of how well the output of a process fits the set of input observations.It is statistics data of the closeness of the data and the fitting regression line. The quality of the fitting will increasewith an increase in the value of the degree of a polynomial. From Table 3, it is observed that increase in the degree ofpolynomial the relationship between SoC and voltage increases fitting accuracy. The regression method is also used topredict the future values of voltage or SoC given that any one of them is given. A polynomial equation of Degree atwhich the value of R-square is maximum (close to 1) can be used as a base parameter to predict variables provided theusage of the battery is not changed. Similarly, RMSE decreases in magnitude for higher order polynomials. In this manuscript, we have investigated the response of the State of Charge (SoC) and the open-circuit voltage acrossthe dynamic battery model under the variable voltage and current during the charging cycle of the battery. Thesevariable input voltage and current have been obtained using the variable irradiance and surface temperature of a SolarPV array which is connected as an input of the dynamic battery model to store the energy within it. In order to matchthe Simulation result with reality, these variable irradiance and surface temperature of Solar PV Array with respectto time has been simulated. After forming and storing the energy within the dynamic battery model; the SoC of thebattery has been estimated using the Kalman filter approach. After the successful estimation of SoC; the Open CircuitVoltage (OCV) and State of Charge (SoC) have been plotted using the polynomial regression technique. The regressionplots between the OCV and SoC have been drawn for the polynomial degree of 2, 3,4, and 5. Results reveal that R keeps increasing as we increase the degrees of regression. Simultaneously the value of RMSE keeps decreasing as weincrease the degree of the polynomial regression. References [1] H. S. Dhiman, D. Deb, and A. M. Foley, “Bilateral gaussian wake model formulation for wind farms: A forecastingbased approach,”
Renewable and Sustainable Energy Reviews , vol. 127, p. 109873, Jul. 2020.[2] H. S. Dhiman, D. Deb, and J. M. Guerrero, “Hybrid machine intelligent SVR variants for wind forecasting andramp events,”
Renewable and Sustainable Energy Reviews , vol. 108, pp. 369–379, Jul. 2019.[3] H. S. Dhiman, D. Deb, and A. M. Foley, “Lidar assisted wake redirection in wind farms: A data driven approach,”
Renewable Energy , Jan. 2020.[4] H. S. Dhiman and D. Deb,
Decision and Control in Hybrid Wind Farms . Springer Singapore, 2020.[5] W.-Y. Chang, “The state of charge estimating methods for battery: A review,”
ISRN Applied Mathematics , vol.2013, pp. 1–7, 2013. [Online]. Available: https://doi.org/10.1155/2013/953792[6] Z. Yu, R. Huai, and L. Xiao, “State-of-charge estimation for lithium-ion batteries using a kalman filterbased on local linearization,”
Energies , vol. 8, no. 8, pp. 7854–7873, Jul. 2015. [Online]. Available:https://doi.org/10.3390/en8087854[7] J. C. A. Anton, P. J. G. Nieto, C. B. Viejo, and J. A. V. Vilan, “Support vector machines used to estimate thebattery state of charge,”
IEEE Transactions on Power Electronics , vol. 28, no. 12, pp. 5919–5926, Dec. 2013.[Online]. Available: https://doi.org/10.1109/tpel.2013.2243918[8] A. Almasoud and H. M. Gandayh, “Future of solar energy in saudi arabia,”
Journal of KingSaud University - Engineering Sciences , vol. 27, no. 2, pp. 153–157, Jul. 2015. [Online]. Available:https://doi.org/10.1016/j.jksues.2014.03.007[9] R. Arshad, S. Tariq, M. U. Niaz, and M. Jamil, “Improvement in solar panel efficiency using solar concentration bysimple mirrors and by cooling,” in . IEEE, Apr. 2014. [Online]. Available: https://doi.org/10.1109/icreate.2014.6828382[10] T. Zhu and L. Wang,
State Energy Transition . Springer Singapore, 2020. [Online]. Available:https://doi.org/10.1007/978-981-32-9499-8[11] M. Glavin and W. Hurley, “Battery management system for solar energy applications,” in
Proceedings ofthe 41st International Universities Power Engineering Conference . IEEE, Sep. 2006. [Online]. Available:https://doi.org/10.1109/upec.2006.367719[12] D. P. Kaundinya, P. Balachandra, and N. Ravindranath, “Grid-connected versus stand-alone energy systems fordecentralized power—a review of literature,”
Renewable and Sustainable Energy Reviews , vol. 13, no. 8, pp.2041–2050, Oct. 2009. [Online]. Available: https://doi.org/10.1016/j.rser.2009.02.00210
PREPRINT - A
UGUST
21, 2020[13] M. Powalla, S. Paetel, D. Hariskos, R. Wuerz, F. Kessler, P. Lechner, W. Wischmann, and T. M. Friedlmeier,“Advances in cost-efficient thin-film photovoltaics based on cu(in, ga)se 2,”
Engineering , vol. 3, no. 4, pp.445–451, Aug. 2017. [Online]. Available: https://doi.org/10.1016/j.eng.2017.04.015[14] V. Vega-Garita, A. Hanif, N. Narayan, L. Ramirez-Elizondo, and P. Bauer, “Selecting a suitable battery technologyfor the photovoltaic battery integrated module,”
Journal of Power Sources , vol. 438, p. 227011, Oct. 2019.[Online]. Available: https://doi.org/10.1016/j.jpowsour.2019.227011[15] K. Abed, A. Bahgat, M. Badr, M. El-Bayoumi, and A. Ragheb, “Experimental study of battery state of chargeeffect on battery charging/discharging performance and battery output power in pv energy system,”
ARPN Journalof Engineering and Applied Sciences , vol. 13, pp. 739–745, 01 2018.[16] L. M. Musanga, W. H. Barasa, and M. Maxwell, “The effect of irradiance and temperature on the performance ofmonocrystalline silicon solar module in kakamega,”
Physical Science International Journal , vol. 19, no. 4, pp.1–9, Oct. 2018. [Online]. Available: https://doi.org/10.9734/psij/2018/44862[17] L. C. and, “Electrochemical model parameter identification of lithium-ion battery with temperature and currentdependence,”
International Journal of Electrochemical Science , pp. 4124–4143, Apr. 2019. [Online]. Available:https://doi.org/10.20964/2019.05.05[18] C. Lyu, Y. Zhao, W. Luo, and L. Wang, “Aging mechanism analysis and its impact on capacity loss of lithium ionbatteries,” in . IEEE, Jun. 2019.[Online]. Available: https://doi.org/10.1109/iciea.2019.8833827[19] L. Zhang, H. Peng, Z. Ning, Z. Mu, and C. Sun, “Comparative research on RC equivalent circuit models forlithium-ion batteries of electric vehicles,”
Applied Sciences , vol. 7, no. 10, p. 1002, Sep. 2017. [Online]. Available:https://doi.org/10.3390/app7101002[20] D. Rosewater, S. Ferreira, D. Schoenwald, J. Hawkins, and S. Santoso, “Battery energy storage state-of-chargeforecasting: Models, optimization, and accuracy,”