Maximizing Charging Efficiency of Lithium-Ion and Lead-Acid Batteries Using Optimal Control Theory

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1 25 American Control Conference Palmer House Hilton July Chicago IL USA Maximizing Charging Efficiency of Lithium-Ion an Lea-Aci Batteries Using Optimal Control Theory Yasha Parvini an Aralan Vahii Abstract Optimal charging of stan-alone lea-aci an lithium-ion batteries is stuie in this paper. The objective is to maximize the charging efficiency. In the lithium-ion case two scenarios are stuie. First only electronic resistance is consiere an in the next step the effect of polarization resistance is also inclue. By consiering constant moel parameters for the lithium-ion battery analytical solutions exists for both scenarios using Pontryagins minimum principle. In lea-aci chemistry the variation of total internal resistance with state of charge () is consierable an the optimal charging problem results in a set of two nonlinear ifferential equations with one initial an a final conition to be satisfie. This so calle two point bounary value problem is solve numerically. I. INTRODUCTION Batteries have become an inispensable part of our aily life. They can be foun everywhere powering our electronic gagets computers an phones; they have mae possible electrifying the transportation sector hybri an electric cars an form a critical part of moern energy gris with renewable energy sources. The nee for increase energy an power ensity an longer cycle life has spurre much research an evelopment towars more efficient batteries an has also calle for more effective battery management systems that monitor an control cell voltages an temperatures. Despite the recent evelopments the limitations in power ensity an also performance reuction at low (high) temperatures still exist ue to high internal resistance of batteries. A major performance bottleneck in stan-alone or hybriize battery systems is ue to resistive losses uring charge an ischarge cycles. When hybriize (e.g. using supercapacitors []) there are extra egree(s) of freeom for shaping the battery charge an ischarge profile an the energy management strategy can be esigne with the goal of reucing resistive losses an increasing overall system efficiency. In stan-alone operation an uring ischarge the cycle is often impose by the require loa an therefore there is little that can be one in reucing resistive losses. During charging however there is the opportunity to choose the charging time an profile such that resistive losses are reuce. Battery manufacturers often have a recommene charging profile which may be sub-optimal. Optimal charging of lithium-ion batteries is stuie in [2] which has focuse on minimizing the charging time while satisfying specific physical an thermal constraints. Yasha Parvini an Aralan Vahii are with the Department of Mechanical Engineering at Clemson University Clemson SC USA sparvin@clemson.eu avahii@clemson.eu In [3] the focus has been on optimal charging of stanalone supercapacitors an uring regenerative braking by minimizing ohmic losses. Suthar et al. in [4] use a singleparticle moel an aim to fin the optimal current profile with the objective of maximizing the charge store in the cell in a given time an constraints of minimal amage to the electroe particles uring intercalation. Bashash et al. in [5] focus on optimizing the timing an rate of charging of a plug-in hybri electric vehicle from the power gri where the goal is to simultaneously minimize the total cost of fuel an electricity an the total battery health egraation. Optimizing the battery charging power in photovoltaic battery systems is stuie in [6] where ifferent objectives such as charging time battery life time an cost of charging are consiere. The equivalent electric circuit epicte in Fig. is use to represent the electrical moel of the battery in this paper. In this figure OCV an V T represent the open circuit voltage an terminal voltage of the battery respectively. In this moel R s inicates the ionic an electronic resistance of electrolyte an also the electronic resistance of the electroe. Charge-transfer resistance R is in parallel with the ouble layer capacitance C which is forme at the interface between the electroe an electrolyte [7]. At steay state the sum of R s an R is calle the total internal resistance (R) in this paper. In this stuy the objective is to maximize the charging efficiency by minimizing the resistive losses in a given charging time an a specifie range of the battery. The charging event is assume to be conucte in a constant ambient temperature. The charging times of interest are the ones that meet a stanar charging metho recommene by the manufacturer. The reason is that in such charging conitions the change in temperature of the cell an also the physical stresses are minimum. This allows to assume temperature inepenence of the moel parameters an also neglect the possibilities of thermal runaways. Fig.. Schematic of the single RC moel /$3. 25 AACC 37

2 The optimal charging problem for the lithium-ion battery is formulate in two steps. In the first step only R s is consiere. With the stanar charging assumption the epenence of R s on temperature is negligible. The epenence of R s on is also shown to be negligible. In the secon step the R-C branch is ae to the moel an the optimal control problem is solve by taking into account the effect of charge transfer resistance an ouble layer capacitance. The simplifying assumption of constant R an C results is analytical solution for the optimal charging current. This result coul be use as a first step for researchers when approaching the problem by consiering the epenence of moel parameters on an even temperature. The optimal charging problem for the lea-aci battery is formulate similar to the first scenario in the lithiumion battery except that the total internal resistance (R) is moele. The efficiency maximization problem is solve by consiering the epenence of the total internal resistance on. This problem structure results in a two point bounary value problem with two nonlinear ifferential equations. Numerical methos are use to solve this problem. II. OPTIMAL CHARGING FORMULATION OF THE BATTERY The objective of this optimal control problem for both battery chemistries is to maximize the charging efficiency by minimizing the ohmic losses. The battery is an inicator of the amount of charge store in the battery at each time normalize by the maximum acceptable charge. The ynamics of the as the common state x (t) for both lithium-ion an lea-aci batteries is erive by performing coulomb counting using the current fe into the battery. Consiering the charging current as the single input u(t) to the system the state equation is governe by the following ifferential equation: t x (t) = u(t) () where is the nominal battery capacity. The optimal charging problem formulation for lithium-ion an lea-aci batteries is escribe in the following sections. A. Optimal Charging of the Lithium-Ion Battery The lithium-ion battery use in this stuy represents the LiFePO4 chemistry. The cell (ANR2665) has a nominal voltage an capacity of 3.3 V an 2.5 Ah respectively [8]. ) First scenario: In this scenario only R s as epicte in Fig. is consiere. The value of R s is constant an equal to.ω. This value is obtaine at 25 C by parameterizing the equivalent electric circuit moel of the cell using pulse-relaxation tests an minimizing the least square error between the experimental an moele terminal voltages [9]. The cost function to be minimize is the ohmic losses associate with R s uring the given charging time t f governe by the following equation: J = R s u(t) 2 t (2) Following the variational approach in optimal control an utilizing the Pontryagin s minimum principle the Hamiltonian is forme as follows: H(xut) = R s u(t) 2 + λ (t) u(t) (3) where λ (t) is the Lagrange multiplier or the co-state. The necessary conitions of optimality shoul be satisfie as follows: H = λ (t) x t H u = (4) Knowing that an R s are constant parameters the Hamiltonian in equation (3) will only be a function of the system input an therefore H x will be zero in equation (4). This inicates that the erivative of the co-state with respect to time is zero an the co-state shoul be a constant. Taking the erivative of the Hamiltonian with respect to the input an setting it to zero accoring to equation (4) results in the following optimal charging current: u (t) = λ (5) 2 R s where enotes the optimal solution. The parameters λ R s an are all constants in equation (5) which inicates that the resulting optimal charging current is also constant. Integrating equation () with the knowlege of u(t) being constant an using the bounary conitions x () = i an x (t f ) = f the value of this optimal an constant charging current is erive as follows: u (t) = ( f i ) t f (6) This is in fact the minimizing solution since 2 H u 2 = 2R s >. Given a specific charging time the most efficient way to charge the battery will be applying a constant current equal to equation (6). For example the optimal strategy of charging the battery from zero charge to full charge in one hour is to apply a constant current equal to 2.5 A. Smaller constant currents compare to the optimal constant current result in lower resistive losses but will not meet the require charging time. On the other han higher constant currents compare to the optimal constant current result in faster charging but with higher resistive losses. The stanar charging metho recommene by the manufacturer is to charge the battery with a constant current-constant voltage (CC-CV) protocol at a rate of C (2.5A). 2) Secon scenario: In this scenario the R-C branch is ae to the moel to inclue the effect of the polarization resistance R. The value of R is assume to be constant an not a function of temperature or. The value for R is.6ω which is an average value over the range [9]. Assume I an I 2 are the currents passing through R an C respectively. Applying the Kirchoff s current an voltage laws to the R-C branch the secon state equation governing the ynamics of the current passing through R is obtaine. 38

3 The problem in this case is to obtain the optimal charging current for a secon orer system governe by the following state equations: t x (t) = u(t) t x 2(t) = R C [u(t) x 2 (t)] (7) where the two states x an x 2 are the of the battery an the current passing through the polarization resistance R. The objective similar to the first scenario is to maximize the charging efficiency. The ifference is that the contribution of the polarization resistance to the total ohmic losses is also consiere. The cost function to be minimize is: J 2 = [R s u(t) 2 + R x 2 (t) 2 ]t (8) The Hamiltonian in this case is given by the following equation: H(xut) = R s u(t) 2 + R x 2 (t) 2 + λ (t) u(t) + λ 2(t) R C [u(t) x 2 (t)] (9) The necessary conitions for optimality are: H = λ (t) H = λ 2(t) x t x 2 t H u = () From the first two conitions the ynamics of the co-states are erive an from the thir conition the optimal input is obtaine as follows: u (t) = ( )λ (t) + ( )λ 2 (t) () 2R s 2R s R C Substituting the optimal input into the state equations in (7) the optimal state ynamics are erive. The result is the following set of four linear first orer orinary ifferential equations (ODE): t x (t) = a λ (t) + a 2 λ 2 (t) t x 2(t) = b x 2 (t) + b 2 λ (t) + b 3 λ 2 (t) (2) t λ (t) = t λ 2(t) = c x 2 (t) + c 2 λ 2 (t) where a a 2 b b 2 b 3 c an c 2 are constant parameters equal to: a = 2R s q 2 max a 2 = b = R C b 2 = b 3 = 2R s R 2 C2 2R s R C 2R s R C c = 2R c 2 = R C (3) Solving this system of linear ODE s simultaneously results in four algebraic equations with four unknowns. The unknown constants are obtaine by applying the bounary conitions specific to this problem which consist of two initial an two final conitions. The initial an final conition for x = are: x (t ) = i x (t f ) = f (4) where i an f are specifie accoring to the esire range of charging. In this specific problem the charging time is specifie an fixe while the values of the secon state at the initial an final time are free; this results in the following equations for the remaining two bounary conitions []: h x 2 (x 2 (t )) = λ 2 (t ) = Initial conition for x 2 h x 2 (x 2 (t f )) = λ 2 (t f ) = Final conition for x 2 (5) In general h(x(t f )t f ) is the term involving the final states an final time in the cost function which in this stuy is zero. Given all bounary conitions one can solve for the states co-states an the optimal input. Consier charging a battery cell from zero charge i = x () = to full charge f = x (t f ) = in one hour. The results for this example are epicte in Fig. 2. I (A) Optimal Current (A) Time (s) Fig. 2. Optimal charging current I an for charging the lithium-ion battery from zero to full charge in hour The optimal charging current for this scenario is slightly ifferent from the result of the first scenario. The optimal 39

4 input in this case is almost a constant current equal to 2.5 A in the majority of times. It may be insightful to also show the result for a fast charging case. Fig. 3 shows the optimal charging current an the two states of the system when the cell is charge from zero to full charge in 5 minutes Full Range.<<.95 Linear Fitting Optimal Current (A) OCV (V) The portion of the OCV profile use for the linear fitting Fig. 4. OCV versus for the lithium-ion battery 2 I (A) Time (s) Fig. 3. Optimal charging current I an for charging the lithium-ion battery from zero to full charge in 5 minutes This charging strategy is not practical ue to thermal an physical constraints plus safety an battery egraation problems. Although it may be interesting to observe that by reucing the charging time the optimal profile iffers from the constant current result observe in the first scenario an also slow charging in the secon scenario. B. Lithium-Ion Efficiency Analysis The open circuit voltage (OCV) of the lithium-ion battery use in this stuy as a function of is epicte in Fig. 4 [9]. As shown in the figure the OCV can be approximate by a linear function using the OCV ata in the range of to 95 percent. This linear function fitting an also the assumption of a constant total internal resistance (R = R s + R =.26Ω) makes the analytical efficiency analysis possible. The linear approximation of the OCV is governe by: V (t) = a(t) + b (6) where V (t) is the open circuit voltage of the battery. In orer to fin the optimal charging efficiency the total energy store in the battery an energy loss is require. The efficiency is then: ρ = E Battery E Battery + E Loss (7) The energy loss in the battery uring optimal charging is alreay known an is equal to Ru (t)2 t where R is the total internal resistance. The total energy store in the battery is: E Battery = = = q f q i V (t)i(t)t V (t) q(t) t t V (t)q (8) where I(t) V (t) an q(t) are the battery current OCV an the charge in ampere-hours respectively. The relationship between OCV an the charge store in the battery is obtaine by consiering the fact that: (t) = q(t) (9) where is the nominal capacity of the battery in amperehours (Ah). Substituting for in equation (6) from equation (9) the linear relationship between V (t) an q(t) is obtaine as follows: V (t) = a q(t) + b (2) where a an b for this specific battery are.56 an respectively. The nominal capacity of the battery is 2.5 Ah. Performing the integration by substituting V (t) from equation (2) into equation (8) an using the efinition in equation (9) for the initial an final the maximum energy store in the battery is obtaine as follows: 32

5 E Battery = ( f i )[ a 2 ( f + i ) + b)] (2) where the unit for energy is watt-hours (Wh). The real maximum amount of energy which the battery can store is obtaine from integrating the original OCV q profile which results in 8.2 Wh for the cell use in this stuy. Using equation (2) an charging the battery from zero charge to full charge the maximum energy store in the battery is calculate as 8.26 Wh. This illustrates that the linear approximation of OCV for lithium ion battery is an effective approach to perform analytical efficiency analysis. Substituting the expressions for E loss an E battery in equation (7) the optimal charging efficiency of lithium-ion battery is obtaine as follows: ρ = + R( f i ) t f ( 2 a( f + i )+b) (22) where t f is the charging time in hours. Consier charging the lithium-ion battery from some initial charge i to full charge ( f = ) then Fig. 5 shows the optimal efficiency as a function of t f /R an for four ifferent initial s. The result shows that starting the charging from a higher initial results in better charging efficiency. Charging Efficiency (%) i = i =.2 i =.5 i = t f /R Fig. 5. Optimal efficiency versus t f /R for ifferent initial in lithium-ion battery C. Optimal Charging of the Lea-Aci Battery The lea-aci battery use in this stuy is a AP-222EV- NB moule with nominal voltage an capacity of 2 volts an 22 Ah respectively []. The real capacity of the moule is 9.7 Ah which is obtaine by ischarging the fully charge moule with a small current of.55a from the upper to the lower voltage limit. Similar to the lithium-ion battery specifically esigne pulse-relaxation tests such as the metho use in [2] is utilize to erive the total internal resistance of the cell (R) as a function of. Fig. 6 shows that the total internal resistance for a lea-aci battery is strongly epenent on. R (Ohm) Fig. 6. Total internal resistance versus for the lea-aci battery The lea-aci battery is moele by a single internal resistance an the only state is the state of charge of the battery governe by equation (). The optimal control is subject to minimize the losses associate with the total internal resistance. Therefore the Hamiltonian is: H(xut) = R(x )u(t) 2 + λ 3 (t) u(t) (23) where R is the total internal resistance an λ 3 (t) is the co-state. Here R(x ) is approximate by a secon orer polynomial function of the state x as follows: R(x ) =.98x 2.2x +.6 (24) The necessary conitions to be satisfie are: H = R(x ) u 2 (t) = λ 3(t) x x t (25) H u = 2R(x )u(t) + λ 3(t) = (26) Solving for u(t) in equation (26) the optimal charging current is obtaine as follows: u (t) = 2 R(x ) λ 3(t) (27) Substituting u (t) from equation (27) in equations () an (25) the following set of two couple nonlinear ODEs are obtaine: λ 3 (t) t x (t) t = 2q 2 max R(x ) λ 3(t) (28) = 4q 2 max R(x ) x R 2 (x ) λ 2 3 (t) (29) Charging the lea-aci battery in t f units of time from zero to full charge requires the following initial an final conitions to be satisfie: 32

6 x() = i x(t f ) = f (3) The system of two nonlinear ODEs with one initial an another final conition forms a two point bounary value problem which coul only be solve using numerical methos. The optimal charging current is obtaine by solving for λ 3 (t) an x (t) = (t) an substituting in equation (27). III. NUMERICAL RESULTS In this section the numerical solution for the optimal charging of the lea-aci battery is presente. Optimal charging problem for lea-aci battery was formulate in the previous section. The result was a set of two nonlinear ODEs with one initial an another final conition. One way to solve this system of ODEs is to specify the initial conition for the an iteratively guess the initial conition for λ 3 until reaches the final specifie value. This metho coul be applie by using ODE solvers in Matlab. Consier the case of charging the lea-aci battery moule from zero to full charge in one hour. Fig. 7 shows the variation of optimal charging current an λ 3 by time. λ 3 Optimal Charging Current (A) x Time(s) Fig. 7. Optimal charging current an λ 3 profiles for charging the lea-aci battery from zero to full charge in hour As shown in the numerical results the optimal charging current for lea-aci battery is not a constant current profile similar to the first scenario in lithium-ion batteries. In orer to compare the constant current charging of the lea-aci battery with the optimal charging strategy the energy losses in both methos are calculate. For the case of charging the lea-aci battery from zero to full charge in one hour the energy losses ue to the resistive losses with the optimal charging strategy are 46.8 KJ compare to 48.9 KJ for constant current charging. This is a 5.5% of less energy converte to heat which coul be significant in terms of thermal management of battery packs. IV. CONCLUSION This paper investigate the optimal charging strategy for lithium-ion an lea-aci batteries. Formulating the optimal control problem an utilizing Pontryagin s minimum principle analytical result existe for the lithium-ion battery uner certain assumptions. In the lea-aci battery case the results show that the optimal charging current is not necessarily constant. Constant current charging of leaaci battery results in 5.5% higher thermal heating which coul be consiere in thermal management stuies of leaaci batteries. The simplifying assumptions mae in this stuy sets the groun for stuies on the battery optimal charging problem in the future. One irection that we will pursue is applying appropriate thermal constraints to meet the challenges in problems such as fast charging where the temperature variation an its effect on moel parameters plays a significant role. REFERENCES [] Y. Parvini J. B. Siegel A. G. Stefanopoulou an A. Vahii Preliminary results on ientification of an electro-thermal moel for low temperature an high power operation of cylinrical ouble layer ultracapacitors in American Control Conference (ACC) 24. IEEE 24 pp [2] R. Klein N. Chaturvei J. Christensen J. Ahme R. Fineisen an A. Kojic Optimal charging strategies in lithium-ion battery in American Control Conference (ACC) 2 June 2 pp [3] Y. Parvini an A. Vahii Optimal charging of ultracapacitors uring regenerative braking in Electric Vehicle Conference (IEVC) 22 IEEE International. IEEE 22 pp. 6. [4] B. Suthar V. Ramaesigan S. De R. D. Braatz an V. R. Subramanian Optimal charging profiles for mechanically constraine lithiumion batteries Physical Chemistry Chemical Physics vol. 6 no. pp [5] S. Bashash S. J. Moura J. C. Forman an H. K. Fathy Plug-in hybri electric vehicle charge pattern optimization for energy cost an battery longevity Journal of Power Sources vol. 96 no. pp [6] J. Li an M. A. Danzer Optimal charge control strategies for stationary photovoltaic battery systems Journal of Power Sources vol. 258 pp [7] S. Sato an A. Kawamura A new estimation metho of state of charge using terminal voltage an internal resistance for lea aci battery in Power Conversion Conference 22. PCC-Osaka 22. Proceeings of the vol. 2. IEEE 22 pp [8] Nanophosphate High Power LithiumIon Cell ANR2665M-B A23 Systems. [9] X. Lin H. E. Perez S. Mohan J. B. Siegel A. G. Stefanopoulou Y. Ding an M. P. Castanier A lumpe-parameter electro-thermal moel for cylinrical batteries Journal of Power Sources vol. 257 pp. 24. [] D. Kirk Optimal Control Theory: An Introuction. Mineola New York: Dover Publishers 24. [] AP-222EV Technical Specifications Amstron Power Solutions. [2] S. Fiorenti J. Guanetti Y. Guezennec an S. Onori Moeling an experimental valiation of a hybriize energy storage system for automotive applications Journal of Power Sources vol. 24 pp