Non-ideal Linear Operation Model for a Li-ion Battery

Non-ideal Linear Operation Model for Li-ion Batteries Alv aro Gonzalez-Castellanos , Student Member , IEEE , David Pozo , Senior Member , IEEE , Aldo Bischi Abstract —Currently , the characterization of electric energy storage units used for po wer system operation and planning models r elies on tw o major assumptions: charge and discharge efficiencies, and power limits are constant and independent of the electric energy storage state of charge. This approach can misestimate the av ailable storage flexibility . This work proposes a detailed model for the characterization of steady-state operation of Li-ion batteries in optimization prob- lems. The model characterizes the battery perf ormance, including non-linear charge and discharge power limits and efficiencies, as a function of the state of charge and requested po wer . W e then derive a linear ref ormulation of the model without introducing binary v ariables, which achieves high computational efficiency , while providing high approximation accuracy . The proposed model characterizes more accurately the performance and technical operational limits associated with Li-ion batteries than those present in classical ideal models. The developed battery model has been compar ed with three modelling appr oaches: the complete non-con vex formulation; an ideal model typically used in the power system community; and a mixed integer linear reformulation approach. The models ha ve been tested on a network-constrained economic dispatch for a 24- bus system. Based on the simulations, we observed approximately 12% of energy mismatches between schedules that use an ideal model and those that use the model proposed in this study . Index T erms —Li-ion battery , non-ideal energy storage, eco- nomic dispatch, con vex optimization N O M E N C L AT U R E Indexes g Generation unit. l Power line. n Power node. t T ime step. j, k Index es for characterization sample sets J and K . P arameters A k Interaction parameters for Redlich-Kister equation [J · mol − 1 ]. A nl Line-to-node incidence matrix. A SEI Area of solid-electrolyte interface [m 2 ]. C Cost [$/Wh]. χ and/ctd Anode/catode molar fraction. ∆ Size of time step [h]. δ /δ Minimum/Maximum angle allowed [rad]. E A Activ ation energy [kJ · mol − 1 ]. η c Coulombic ef ficiency . η cha/dis Charge/dischar ge efficiency . E Battery energy capacity [Wh]. F Maximum po wer flow [W]. F Faraday constant [s · A · mol − 1 ]. Γ s,n Battery-to-node incidence matrix. i Current [A]. k 0 Reaction rate constant [m · s − 1 ]. Ω g ,n Generator-to-node incidence matrix. P Power [W]. P /P Minimum/maximum po wer [W]. R Gas constant, [J · mol − 1 K − 1 ]. R ct Charge transfer equiv alent resistance [ Ω ]. R dif ,elec/mem Electrode/Membrane diffusion equi valent resistance [ Ω ]. R ohm Ohmic losses equivalent resistance [ Ω ]. S OC State of charge [p.u.]. S OC sur State of charge at electrodes’ surf ace [p.u.]. T T emperature [K]. U bat, 0 Reference equilibrium potential [V]. v eq Equilibrium voltage [V]. v I N T Non-ideal interaction voltage [V]. v I N T ,and/ctd Non-ideal anode/catode interaction voltage [V]. X l Series reactance in the line l [p.u.]. I . I N T RO D U C T I O N T HE need for a secure and flexible operation of electric power systems and the falling prices of batteries has made large-scale Electric Energy Storage (EES) systems a viable and widely studied option. In particular , Li-ion battery systems have gained considerable attention because of their high energy density , po wer ratings, ef ficiency , and long life- time [ 1 ]. As an example, in South Italy and South Australia, 40 MW and 100 MW storage systems hav e been installed, respectiv ely [ 2 ], [ 3 ]. The introduction of EES, alongside re- new able ener gy generation, compels the dev elopment of tools that can manage these systems optimally for maximum grid reliability and profitability [ 4 ]. The two major applications of EES that help achie ve higher profit potential include frequency regulation and load shifting [ 5 ]. Multiple market mechanisms ha ve been proposed for providing services to the power grid from EES: bulk energy storage, vehicle-to-grid (V2G), and distributed battery networks. V2G enables the use of electric vehicles to provide the aforementioned services, in addition to demand shifting via the smart charging of v ehicles [ 6 ]. Domestic batteries can also be used for grid-scale services through their aggreg ation [ 7 ]. Gi ven the fast response needed for primary frequency control applications (in the range of seconds), battery systems would require the modeling of their dynamic processes. On the other hand, load shifting applications assume steady state 1 behavior , i.e., constant parameters during the operational time steps considered, from minutes to hours. Battery models can be di vided into mathematical, electro- chemical, and equi v alent-circuit [ 8 ] models. The first category , mathematical models , describes battery behavior based on the state of charge - S O C (i.e., the ratio between stored ener gy and battery capacity), state of health (the battery’ s ability to perform in comparison to manuf acturer specifications), and other macroscopic properties. Computational methods are used to derive mathematical models with a small number of variables, resulting in low computational costs, but without reflecting the internal processes in the cells, e.g., the changes on the equiv alent voltage as a function of the stored energy . The second category , electr ochemical models , describes the chemical beha vior of battery cells based on the physical and chemical processes that occur in the battery cells. Because of their accuracy , electrochemical models can be applied to the optimization of processes that are related to the design of the cell’ s physical parameters. Additionally , reduced-order electrochemical models allow the accurate modelling of the electrochemical processes in the battery , while providing a computationally efficient storage characterization for online control applications [ 8 ]. Howe ver , dynamic models are not suitable for their use in optimization applications which re- quire the analysis of large time horizons. The third category , equivalent-cir cuit models , pro vides an equi v alent representa- tion of the battery cell based on laboratory measurements. The electric elements in the circuit can be denoted by a combination of linear and exponential models to attune for the dynamic processes in the battery . The resultant circuits provide computationally efficient models, without requiring time-consuming laboratory measurements for the estimation of battery parameters [ 9 ]. V agr opoulos et al. developed a linear optimization model for the optimal operation of an electric vehicle aggregator [ 10 ]. The model is based on battery charging processes at constant efficienc y , which are di vided into two stages: constant current and constant voltage charge. P and ˇ zi ´ c et al., provided piecewise linearized charging limits based on the battery’ s stored ener gy levels [ 11 ]. Their model is based on a constant- current-constant-voltage char ging cycle and v alidated based on laboratory experiments. The effect of the requested current in storage efficiency is modeled by W ang et al. [ 12 ]. A concave and monotonically increasing function is provided to represent battery efficiency in terms of deviations from the system’ s reference current. A detailed characterization of the loss mechanisms prev alent in both battery and power electronics is provided by Schimpe et al. [ 1 ]. An analysis of the efficiency changes as a function of the stored ener gy level and po wer request is provided by Morstyn et al. [ 13 ]. The ener gy storage system characterization is approximated as a second-order model for its application in model predictiv e control for distributed microgrids with photov oltaic generation. Ali et al. de veloped a methodology for the non-linear estimation of Li-ion battery parameters in an equiv alent circuit model [ 9 ]. The estimated parameters reflect the dynamic processes by characterizing the elements in the equiv alent circuit model as a function of the S O C . Berrueta et al. dev eloped an equiv alent circuit model for a Li-ion battery based on experimental data and the underlying electrochemical phenomena that characterize its performance [ 14 ]. The proposed model achieves high accuracy while being computationally simple to implement. Additionally , the model characterizes the battery by the use of its state variables: cell temperature, current, and SOC. An ef ficiency-based equiv alent-circuit model has been proposed by Rampazzo et al. [ 15 ]. The model simulates the battery performance based on the battery state during the operation, as in [ 14 ]. A Mixed Integer Linear Programming (MILP) model for representing the beha vior of a Li-ion battery pack based on the battery’ s electrochemical behavior was developed by Sakti et al. [ 16 ]. In this MILP model, the power limits and battery ef ficiency are expressed as a function of the S O C and the po wer output. The nonlinearities present in the characterization are addressed through piecewise linearization based on simulated sample points. A. P aper Contributions and Or ganization The primary contribution of this study is the dev elopment of a detailed linear Li-ion battery model for its use in the operation optimization of power systems. This work takes into account the use of energy storage for economic oper- ations in which the time steps considered are in the order of minutes/hours. Therefore the transient characterization of the batteries is assumed negligible. The proposed method- ology models the operation limits and efficiencies of Li-ion batteries based on the characteristic charging and discharging curves. These characteristic curves can be obtained based on either computational simulations or by direct measurements, which allo ws the proposed model to be adapted to different energy storage technologies and updated when new simu- lated/measured data is available. For this purpose we de v ote Section II for introducing an equiv- alent circuit representation for the electrochemical behavior of Li-ion batteries, inspired by the work of Berrueta et al. [ 14 ]. In Section III we introduce a detailed non-linear model extended from the described equiv alent circuit for the state- dependent charge, and discharge processes of Li-ion batteries based on the characterization of battery efficiencies and power limits. Thirdly , in Section IV , we propose a linear approach for battery operation through the con ve x curves definition of detailed non-linear battery models. Lastly , Section V and VI integrate our proposed Li-ion battery model into a day- ahead economic dispatch to illustrate the benefits of a detailed linear battery model compared with: the non-conv ex formu- lation of Section II, existing ideal model approaches (see, e.g., [ 17 ]), and approximation of non-con vex curves through piecewise linearization based on mixed-integer programming. The computational and reliability benefits of implementing the dev eloped model are also quantified. I I . B A T T E RY E L E C T R O C H E M I C A L M O D E L A Li-ion battery re versibly stores electric energy as electro- chemical energy . The positiv e electrode, cathode, of a Li-ion 2 battery is composed of metal oxide materials, usually transition metals, and the negati ve electrode, anode, is made of graphite. When the battery is being charged, the Li ions flow from the cathode to the anode, where they combine with the incoming electrons and are later stored in the graphite layers [ 18 ]. During the discharge process, electrons are transferred from the battery cell, and the Li ions flow through the electrolyte back to the cathode. After its dynamic processes hav e stabilized, a Li-ion battery can be represented as an equiv alent resistiv e circuit (Fig. 1). The equiv alent-circuit representation consists of a voltage source associated with the electrochemical equilibrium voltage and three resistors that represent different electrochemical processes: ohmic losses, charge transfer , and membrane dif- fusion. In this section we describe each of the elements and its correspondence with the underlying electrochemical phenomena based on the experimentally validated model of Berrueta et al. [ 14 ]. v eq ( S OC , T ) + − R ohm ( S OC , T ) R ct ( S OC , T ) R dif ,mem ( T ) i + − v Fig. 1. Steady-state battery equiv alent circuit A. Equilibrium voltage The equilibrium voltage, v eq , indicates the difference in electrochemical potential (v oltage) between the electrodes after the charge transfer dynamic processes hav e reached a steady state. Battery equilibrium voltages can be expressed as a summation of the three processes inside the cell, (1). The cell reference potential is defined by U bat, 0 , which indicates the cell potential at standard concentrations. The concentration, i.e., molar fraction χ , of the reactants on the electrodes changes with the amount of stored energy , which is captured by expressions (1a) and (1b). Therefore, there is a change in the cell potential for nonstandard concentrations obtained by the second term in the summation. The combination of the first two terms giv es the Nernst equation for nonstandard conditions (reactions not occurring at 298 . 15 K , 1 atmosphere, or a cathode and anode molarity of 1 . 0 M ). The third term, v I N T , reflects the non-ideal interactions between Li ions and the host matrix. The non-ideal interactions can be calculated based on a Redlich-Kister polynomial equation of se venth order , (1d). v eq ( S OC ,T )= U bat,0 + RT F · ln  (1 − χ ctd ) · χ and χ ctd · (1 − χ and )  + v INT (1) where χ and = 0 . 083 + 0 . 917 · S O C, (1a) χ ctd = 1 − 0 . 7 · S O C, (1b) v INT = v INT ,ctd − v INT ,and , (1c) and v INT ,j = 7 X k =1 A k " (2 χ j − 1) k − 2 χ j ( k − 1)(1 − χ j ) (2 χ j − 1) 2 − k # , (1d) j = and , ctd . B. Resistive elements The resisti ve elements R ohm and R ct indicate fast-dynamic processes. The former is associated with the ohmic phenomena that represents the losses related to the movement of electrons and ions during char ging and dischar ging processes; it depends linearly on the SOC and temperature, (2). The latter, R ct , models char ge transfer through the solid-electrolyte interface (SEI), (3). The SEI serves as a barrier between the electrodes and the electrolyte solution, prev enting their spontaneous re- action (short-circuit) and enabling battery charge reversibility . R ohm ( S OC ,T ) = R ohm,0 + R ohm,T · T + R ohm,SOC · S O C (2) R ct ( S OC ,T )= 1 ( χ α,and · χ α,ctd ) 0 . 5 · " R · T · e E A /R · T F 2 · A S E I · k 0 # (3) The last resistive element of the equiv alent-circuit model is related to the dif fusion of Li ions through the membrane. The diffusion process causes a voltage drop in the battery cell and is inv ersely proportional to the operating temperature (4). Similar to the membrane diffusion process, there exists an electr ode diffusion mechanism (5). R dif ,mem ( T ) = K dif ,mem · exp b dif ,mem T − T 0 ,dif ,mem ! (4) R dif ,elec ( T ) = K dif ,elec · exp b dif ,elec T − T 0 ,dif ,elec ! (5) Changes in the concentration of lithium on the electrodes results in a new perceived state of char ge at the cell’ s elec- trode surface, S OC sur . The difference between the S OC and S OC sur results in a voltage drop that can be obtained in terms of the electrode diffusion process, (6). S OC sur = S OC − R dif ,elec · i · η c (6) where the coulombic efficiency , η c , is given by η c = η c, 0 + η c,T · T + η c,i · i (7) In summary , all of the components present in the equiv alent steady-state circuit model depend either on the temperature, T , the state of charge, S OC , the current, i , or a combination of all factors. For the calculations presented in the following sections, the battery modeled will be that presented in [ 14 ], with 40 Ah and 133 V . A constant battery temperature of 25 ◦ C will be assumed. The S OC and the current will be considered control variables because they directly relate to the stored energy and 3 the power requested from the battery . The values and the characterization of the battery parameters presented in this section, and used in the rest of the work, can be found in [ 14 ]. I I I . M A T H E M A T I C A L B A T T E RY C H A R AC T E R I Z A T I O N In this section, we present an alternativ e formulation to the classical ideal battery model used in the operation optimization of power systems (see e.g., [ 17 ]) by incorporating features that represent the internal electrochemical processes at the battery cell lev el. Consequently , we provide a more accurate descrip- tion of battery behavior that can be used for power system economic operation. In doing so, charging and discharging power and SOC limits must be derived. Howe ver , instead of providing independent limits, as in [ 17 ], we derive the limits from internal battery processes. The resulting mathematical battery model is of higher-dimension (more variables are needed), and it is non-linear and non-con ve x (a non-desired property for power system optimization models). On the other hand, it provides a more accurate mathematical description of the battery that is useful for power system optimization models. In this section, we gradually introduce the model, while in the next section, we propose a con ve x (linear) approach for the battery model. A. P ower Limits The S OC at the electrode surface, S O C sur , is modeled based on (6) and (7) as: S OC sur = S OC − R dif ,elec · i · [ η c, 0 + η c,T · T + η c,i · i ] , (8) where 0 ≤ S OC sur ≤ 1 . (9) During the discharging process, the state of charge at the electrodes’ surface S O C sur decreases; whereas in the charg- ing one, it increases. Therefore the S OC sur lower bound is activ e during the dischar ge (b ut not its upper bound). Similarly , the S OC sur upper bound is activ e during the charge. The calculation of the discharging limits is then derived as follows 0 ≤ S OC cha sur , (10) where the maximum discharging current I dis 0 is the solution of the equation S OC cha sur  I dis 0  = 0 . (11) W e can obtain an explicit solution, in closed form, deriv ed for the maximum discharging and charging current, I dis 0 and I cha 0 , from the second-order polynomial resultant of solving (11). The maximum discharge current can be obtained by solving the follo wing quadratic expression: 0 = a dis ·  I dis 0  2 + b dis · I dis 0 + c dis , (12) with a dis = − R dif ,elec · η c,i , (12a) b dis = − R dif ,elec [ η c, 0 + η c,T · T ] , (12b) c dis = S OC . (12c) Analogously , by bounding the S O C sur in (9) by its upper limit, the maximum charging power can be obtained with the following quadratic expressions: 0 = a cha ·  I cha 0  2 + b cha · I cha 0 + c cha , (13) with a cha = − R dif ,elec · η c,i , (13a) b cha = − R dif ,elec [ η c, 0 + η c,T · T ] , (13b) c cha = S OC − 1 . (13c) The current limits I cha/dis 0 refer to the limitations in the charge transfer process. T o consider the manufacturer limits which prev ent cell damage, the parameter I cha/dis c − rate is introduced. Thus, the maximum permissible current for a battery can be calculated based on I cha/dis 0 and I cha/dis c − rate by: I cha/dis = min { I cha/dis 0 , I cha/dis c − rate } (14) From expressions (12)–(14), it is now possible to calculate the maximum permissible current as a function of the energy stored. The maximum permissible C-rates for the discharge and charge current, I dis/cha c − rate , hav e been set to 5 C 1 and 1 C to match the typical manufacturer limits used in [ 14 ]. Accord- ingly , we have plotted the maximum feasible working points for discharging and charging currents vs. the SOC in Figs 2(a) and 2(b). As shown in Fig. 2(a) for low S O C , the maximum current for discharging decreases. At low S OC , the internal battery resistance, R tot = R ohm + R ct + R dif ,mem , increases con- siderably , decreasing the equi valent voltage, v eq , below zero, which indicates an erroneous sense of battery depletion. This behavior corresponds to the voltage cut-off in the battery cells, which would result in a battery shut down by the management system [ 19 ]. The charging current limits deviate from the 1 C rating for higher values of S O C , which is greater than 0.93, Fig. 2(b). The physical limitations for cell charging correspond to a greater rate of voltage rise, when compared with the rate of charge absorption. This results in the saturation of electrochemical cells and an increase in stresses within the cells. (a) Discharge current (b) Charge current Fig. 2. Maximum current normalized to the battery capacity , as a function of the SOC. Shadow areas indicate feasible operation. 1 A C-rate of 5 C indicates 5 times the nominal current for discharging according to typical manufacturer specifications. 4 The expressions for calculating dischar ging power limits (15a), P dis , and char ging power limits (15b), P cha , are deriv ed using the circuit-equiv alent battery model and currents limits. P dis = v eq · I dis −  I dis  2 · R tot (15a) P cha = v eq · I cha +  I cha  2 · R tot (15b) B. Charging and Dischar ging Battery Efficiencies T o characterize battery usage on an operation optimization model, it is necessary to deri ve an expression for the battery’ s performance, both for charging and discharging regimes. The battery dischar ging efficiency is given by (16). The symbol p dis denotes the po wer discharged to the electric grid, and so p dis = v · i . The symbol p out denotes outgoing power from the battery cells, p out = v eq · i . Applying Kirchoff ’ s V oltage Law on the circuit in Fig. 1, we can derive the discharging efficienc y of the battery . η dis = p dis p out = 1 − i · R tot v eq (16) Similar to discharging efficiency , we can derive a char ging efficiency expression as follows η cha = p in p cha = v eq v eq + i · R tot (17) As seen in Fig. 3(a) and 3(b), the efficiency of a battery improv es for higher S OC and C-rates closer to 1 C . This result corresponds to previous e xperimental analyses performed in [ 15 ], [ 1 ], [ 16 ]. The discharging efficienc y lowers for higher current v alues and low S O C , dropping as much as 33% from its maximum value. The charging efficienc y presents a similar behavior , with a smaller efficienc y drop correspondent to the smaller operating region, 0 − 1 C . (a) Discharge efficiency (b) Charge efficiency Fig. 3. Discharging and charging efficiencies vs. the SOC and discharging and charging power C. Non-Linear Battery Model By considering the aforementioned charging and discharg- ing power limits and efficiencies, we can deriv e a ne w detailed battery model that is similar to con ventional methodologies [ 17 ]. The approach is presented in M O D E L 1 . Equations (18a) and (18b) indicate the discharging and charging power lim- its. Contrary to con ventional models, the limits are SOC- dependent. Equation (18c) denotes the energy balance along the time steps. The ener gy stored in the battery at time t is calculated as the sum of the energy stored in the previous time step t − 1 and the energy entering the battery cell in the previous time-step. It is given by the product of p cha t − 1 η t − 1 ∆ minus the ener gy exiting the cell, which is represented by the term p dis t − 1 1 η dis t − 1 ∆ . The parameter ∆ is the size of time steps as an hourly fraction; employed to transform the use of power into energy . Battery energy capacity limits are described by (18d). The battery state of charge, S O C t , in p.u., is calculated as a function of the stored energy , e t , by (18e). Finally , efficiencies and power limits are included in (18f). M O D E L 1 NLP Li-ion battery model V ariables: p dis t , p cha t − discharging and charging power P dis t , P cha t − maximum discharging and charging power η dis t , η cha t − discharging and charging efficienc y e t , S OC t − battery energy lev el (absolute and relative values) Constraints: 0 ≤ p dis t ≤ P dis t , ∀ t (18a) 0 ≤ p cha t ≤ P cha t , ∀ t (18b) e t = e t − 1 + p cha t − 1 η cha t − 1 ∆ − p dis t − 1 1 η dis t − 1 ∆ , ∀ t (18c) 0 ≤ e t ≤ E , ∀ t (18d) S OC t = e t /E , ∀ t (18e) (15) − (17) (18f) I V . L I N E A R R E F O R M U L A T I O N A P P R O AC H In the aforementioned non-linear M O D E L 1 , non-con vexities arise from discharging and charging power limits and efficien- cies (18f), including bilinear products in the energy balance constraint (18c). In this section, we propose a linear approach through a conv ex env elope for the characterization of battery charge and discharge. T o handle the bilinear products in the battery energy balance equation, we add two new variables, incoming and outgoing power from battery cells, p in and p out , respecti vely . They are introduced in equations (16) and (17) and given by: p out = p dis 1 η dis (19a) p in = p cha η cha (19b) Expressions in (19) allow us to define the energy balance equation as an affine function of p out and p in e t = e t − 1 + p in t − 1 ∆ − p out t − 1 ∆ . (20) This substitution can be done without the need for ev aluating the bilinear products of (19) because η out is dependent on the power provided by the battery , p dis , and the S OC , e.g. a function of i and v eq , see (16). Therefore, the values of p out can be obtained in terms of p dis and S OC , which are the variables of interest in operation models implementing electric energy storage through batteries. 5 The outgoing power from the battery cells, p out , can be approximated by a con v ex combination of sampling points. That is, we can construct a polyhedral en velope of the p out by sampling from the model giv en in Section II, or alterna- tiv ely , by experiments like in [ 14 ]. Therefore, for ev ery tuple [ p dis , S OC , p out ] > , we draw J samples through simulations, represented by [ b P dis j , [ S OC j , b P out j ] > . The linear con vex env e- lope is formulated in (21). A 3-dimensional representation of p dis vs. S O C and p out is presented in Fig. 4(a), where the solid areas indicate the simulated values based on Eq. (19a). The sampled points j ∈ J are highlighted as black dots, and the con vex en velope connecting the sampled points is shown by the connecting lines between the points. As it can be observed, the proposed approach based on the con vex en velope for a feasible set points is very close to the non-linear mathematical definition. p out = X j b P out j · x j (21a) p dis = X j b P dis j · x j (21b) S OC = X j [ S OC j · x j (21c) 1 = X j x j (21d) 0 ≤ x j , ∀ j ∈ J (21e) Analogously , p in can be approached by the con vex env elope defined in (22). Similarly , it is also compared with the non- linear definition in Fig. 4(b). p in = X k b P in k y k (22a) p cha = X k b P cha k y k (22b) S OC = X k [ S OC k y k (22c) 1 = X k y k (22d) 0 ≤ y k , ∀ k ∈ K (22e) The expressions for battery characterization are given as a function of the S O C as a consequence of the dependence of the v eq and power limits on it, including the computational advantages of employing a normalized parameter . The resul- tant linear model for battery characterization is presented in M O D E L 2 . The S O C is characterized by (21c) and (22c). If both expressions were jointly considered, the battery would ap- pear to be charging and discharging at the same time. This corresponds to the fact that to guarantee an approximation through a con ve x combination of sampling points, constraints (21d) and (22d) require that at least one x j and one y j be greater than zero, simultaneously making p dis and p cha non zero. In M O D E L 1 , this does not occur because the energy balance, (18c), uses efficiencies η dis and η cha that are lower than one. Therefore, simultaneously charging and discharging would go against the economic objectiv e of minimizing the (a) Discharging power (b) Charging power Fig. 4. Operating region of Li-ion battery in variable space of (a) [ p dis , S O C , p out ] > , and (b) [ p cha , S O C , p in ] > . The curve indicates non-linear dependence, black dots denote sampled points, and the lines between the sampled points define the conv ex env elope of the sampled points. cost of power system operations because ener gy would be lost during the imperfect (and simultaneous) charge and dischar ge processes. 2 Based on con vex combination constraints and the economic use of the battery , constraints (21c) and (22c) can be combined through a summation in (23f). For this constraint to allow a discernment of the charging and discharging processes without the introduction of binary variables, an additional condition is introduced: each sampling set must hav e at least two sampling points equal to [0 , 0 , 0] > and [1 , 0 , 0] > ; which respectiv ely represent the cases when the battery is not activ e p = 0 , but is fully discharged or fully charged. Consequently , for a discharge p dis > 0 , x j 0 = 0 and then p cha = 0 , since y j 0 = 1 ; an analogous relationship would follow for the charging cycle. For the considered dischar ging con vex approximation by our proposed method we hav e a maximum approximation error of 9.03%, a mean error of 1.21%, and a standard deviation of 1.39%, when 14 sampling points are considered. For the charging process sampled with 20 points, we have a maximum approximation error is of 1.12%, with a mean error of 0.22% and a standard deviation of 0.18%. The greater approximation error for the discharging curve can be explained by the greater 2 This behavior is ensured for a majority of power system applications where the po wer balance constraints can be satisfied within the technical limits of the generators and demand, i.e., when the demand can be fulfilled without recurring load shedding or generation curtailment. 6 M O D E L 2 Linear Li-ion battery model V ariables: e t , S OC t − battery energy lev el (absolute and relative values) p dis t , p cha t − discharging and charging power p out t , p in t − power outgoing and incoming at the cells x j t , y kt − variables related to the sample sets J and K Constraints: e t = e t − 1 +  p in t − 1 − p out t − 1  ∆ , ∀ t (23a) p out t = X j b P out j t x j t , ∀ t (23b) p dis t = X j b P dis j t x j t , ∀ t (23c) p in = X k b P in kt y kt , ∀ t (23d) p cha = X k b P cha kt y kt , ∀ t (23e) S OC t = X j [ S OC j t x j t + X k [ S OC kt y kt , ∀ t (23f) S OC t = e t /E , ∀ t (23g) 1 = X j x j t , ∀ t (23h) 1 = X k y kt , ∀ t (23i) 0 ≤ x j t , ∀ j, t (23j) 0 ≤ y kt , ∀ k , t (23k) changes in current that the sampling points in the y-axis (6 points for both the charge and discharge) must approximate, when compared to the charging range. V . N E T W O R K - C O N S T R A I N E D E C O N O M I C D I S PA T C H W I T H E N E R G Y S T O R AG E D E V I C E S A conv entional economic dispatch, based on a lossless DC approximation, with linear costs is modeled by M O D E L 3 . The scheduled cost of energy generation is giv en by (24). Equation (24a) represents the power balance at ev ery node n of the system for every time step. The power entering the node from each connected line, f lt , is equal to the nodal demand, P D nt , minus the po wer generated at the node, p g t , minus the discharging power of the battery connected to this node, p dis st , plus its charging power , p cha st . The power flowing in line l is modeled by (24b) using line-to-node incidence matrix, A nl . Equations (24c), (24d), and (24e) establish the technical limits of the generators g , po wer lines l and voltage angle at node n , respectively . The reference voltage angle is set by (24f). Battery energy lev el at the end of a dispatch horizon is set to be equal to the initial battery energy lev el (24g). V I . T E S T C A S E W e ev aluated the effect of the proposed characterization on battery operation for electric economic dispatch. F or this purpose, four model approaches were compared: M O D E L 3 Network-Constrained Economic Dispatch with EES V ariables: p g t generated po wer by g during t f lt line po wer flow through l on t δ nt voltage phase angle at n on t Objective: minimize X g ,t C g p g ,t ∆ (24) Constraints: X l A nl f lt = P D nt − X g Ω g n · p g t − X s Γ sn [ p dis st − p cha st ] , ∀ n, t (24a) f lt = 1 X l X n A nl · δ nt , ∀ l (24b) P g ≤ p g t ≤ P g , ∀ i, t (24c) − F l ≤ f lt ≤ F l , ∀ l (24d) δ n ≤ δ n,t ≤ δ n , ∀ n (24e) δ n = n 0 ,t = 0 , ∀ t (24f) e t =1 = e t = | T | (24g) Battery model ( M O D E L 2 ): (23) . (24h) • C A S E N L P : the storage systems is modeled as described in M O D E L 1 , resulting a non-linear non-con ve x (NLP) problem. • C A S E L P – I D E A L : an ideal battery was considered, i.e., M O D E L 1 is employed, but the parameters P cha , P dis , η cha t , and η dis t were assumed to be constant. • C A S E L P – A P P ROX : the battery is modeled as described in M O D E L 2 , employing a con ve x piece wise linearization of its characteristic curve. • C A S E M I L P : for the representation of the non-conv ex nature of the characteristic curves, Fig. 4, a piecewise linearization using binary variables is employed [ 20 ]. The resultant model is a mixed-integer linear programming (MILP) one. The computational tests were performed using a modified version of the IEEE Reliability T est System (IEEE R TS) [ 21 ]. The generator technical data, the line parameters, the load profile, including the battery parameters, are given in the online dataset [ 22 ]. The optimization horizon was set to 24 hours, with 10-minute intervals. The load data was scaled to the IEEE R TS system based on the demand data from the Iberian Electricity Market on June 20, 2018 [ 23 ]. The simulations are performed using the modeling software Julia 0.6.4 [ 24 ], JuMP 0.18 [ 25 ], and Gurobi 7.5.1. [ 26 ] and IPOpt [ 27 ] as solvers. A summary of the characteristics and results of the evaluated cases is provided in T able I. A. Results 1) C A S E N L P - Non-linear pr ogr amming model: The gen- eral non-linear programming (NLP) model, M O D E L 1 , has been ev aluated to provide a reference for the battery operation. The scheduled battery operation is presented for both char ge and discharge in Fig. 5(a) and Fig. 5(b), respectiv ely . The real maximum charging and discharging po wer limits hav e been calculated based on (15a)–(15b) and represented by the red 7 T ABLE I S U MM ARY O F O PT IM I Z ATI O N R ES U LTS NLP Ideal MILP PWL Objectiv e value 57 305 57 312 57 307 57 308 ∆ [%] - 0.01 0.01 0.01 T ime [s] 231.3 1.2 143.3 3.4 Constraints 10 657 9 937 24 049 10 369 Continuous v ariables 18 576 11 088 20 592 16 272 Binary v ariables - - 13 824 - dashed line in the figures. Battery usage follows charging cycles for periods of lo wer demand, to subsequently discharge during higher load request. Power limits for both charge and discharge change throughout the day as a function of the stored energy , as described in Section III-A. 2) C A S E L P – I D E A L – Ideal battery model: In this case, the battery model is based on [ 17 ]. For doing so, we considered M O D E L 1 , where po wer limits and ef ficiencies hav e constant values. The current limits, (14), are set to 1C and 5C. The voltage at the battery terminals is set constant at the rated value of 133 V . For the ideal battery model, the ef ficiencies are set to constant values, η cha t = 0 . 972 and η dis t = 0 . 868 . The constant ef ficiencies were calculated as the mean of the values giv en by (16) and (17). As a result of the economic dispatch defined in M O D E L 3 , the battery was scheduled to charge from and discharge to the network as sho wn in Fig. 5(c) and 5(d), respectively . As it can be seen, the scheduled operation of the ideal battery violates the calculated limits during peak charge and discharge. 3) C A S E L P – A P P R OX - Pr oposed linear battery model: The scheduled operation of the proposed model is given in Fig. 6(a) and Fig. 6(b), characterized by its higher use, i.e., during more time steps. F or this battery model, the ef ficiencies depend on the state of charge and the power request, i.e., higher efficienc y for higher SOC and lower power charge/dischar ge. Consequently , the battery use in this case must balance the power deliv ered with the ef ficiency and the system’ s marginal cost of generation. For this purpose, the battery is used at lower power lev els, when compared to the ideal model with constant efficienc y , to operate at higher efficienc y . This change in scheduling can be observed in more time steps used for charge and discharge. Greater use of the non-ideal battery allows the objective value of this test case to be the same as that obtained for the ideal battery , even though the battery has a varying ef ficiency . 4) C A S E M I L P - Mixed-integ er linear pro gramming model: W ith the aim of comparing the proposed con vex model with one that captures the non con vexity nature of the battery characteristic curves, a mix ed-integer programming model (MILP) has been used to represent the non conv ex piecewise linearization of the curve through the triangle method [ 20 ]. The charging and dischar ging curves ha ve been each represented with 15 sampling points, taking a base of 5 points for the SOC- and 3 for the p out/cha -axis. The higher number of points in the SOC-axis allo ws a better representation of the po wer limits as a function of the stored energy . The results of the MILP model are presented in Fig. 6(c) and Fig. 6(d). This case employs the battery in two charging and (a) C A S E N L P : charging power vs. time (b) C A S E N L P : discharging power vs. time (c) C A S E L P – I D E A L : charging power vs. time (d) C A S E L P – I D E A L : discharging power vs. time Fig. 5. Scheduling of the battery char ge and discharge processes for (a)-(b) C A S E N L P , and (c)-(d) C AS E L P – I D E A L . The dotted line denotes the maximum power as a function of the S O C based on M O D E L 1 . The red areas indicate infeasible operation. discharging cycle with greater intensity , shorter time of use and higher requested power . The objecti v e value resultant of 8 the MILP model is the closest one, albeit by a small margin, to that provided by the complete non-conv ex formulation (NLP). Nonetheless, its computational cost is considerably higher than that of the proposed model, correspondent to the exponential increase of its solution time in relation to the number of binary variables. This makes it unsuitable for real life applications where the amount of batteries and size of the time horizon of interest is considerably larger . B. Reliability Model Assessment The schedule of an ideal battery model for power charging and discharging is possible in unfeasible regions of battery operation. Therefore, it is expected that there will be situations in which the battery cannot provide the power/ener gy required by the schedule. T o calculate this mismatch, we introduce a reliability metric obtained as follows: 1) For the time steps in which p cha t or p dis t consider values outside of the regions defined by (21)–(22), we set their values to be equal to the maximum power attainable for the gi ven S OC . 2) The realized (corrected) charging and discharging sched- ule for the battery is used for updating new ener gy lev els, e real t , based on (18c). 3) The energy imbalance/deviation is then calculated as the sum of the differences between the scheduled and the realized energy lev els, as follows: Imbalance = X t  e real t − e t  (25) Fig. 7 presents the scheduled S OC for an ideal battery and the realized le vels resulting from this analysis. It can be observed that there exists not only a discrepancy between the scheduled and the realized values, but the battery would also reach negati ve energy lev els, i.e., selling more energy than the available one. The energy deviation calculated by (25) accounts for 12 . 2% of the scheduled energy to be stored in the battery , resulting not only in a profit detriment for the owner but also on an ov erestimation of the system reliability that leads to a false sense of flexibility . V I I . C O N C L U S I O N S This paper proposes a detailed model for battery charac- terization in optimization problems. The model is built based on a Li-ion battery , where the relationship between the state of charge, the charging and discharging efficiencies, and the power limits are described. The proposed model is based on a non-linear equiv alent circuit model. A linear reformulation is then proposed based on a con vex en velope for all feasible operation set-points. The linear reformulation is a sample- based approach but very close to the original non-con ve x and non-linear mathematical model. The linear model enables the optimal management of a battery system without the use of binary v ariables. The proposed model has been integrated into an economic dispatch and tested against an ideal battery model, with constant po wer limits and ef ficiencies, that has been commonly used in the power systems literature. A case study for compu- tational tests was performed based on the 24-bus IEEE-R TS (a) C A S E L P – A P P ROX : charging po wer vs. time (b) C A S E L P – A P P ROX : discharging po wer vs. time (c) C A S E M I L P : charging power vs. time (d) C A S E M I L P : discharging power vs. time Fig. 6. Scheduling of the battery char ge and discharge processes for (a)-(b) C A S E L P – A P P R OX , and (c)-(d) C A S E M I L P . The dotted line denotes the maximum power as a function of the S OC based on M O D EL 1 . The red areas indicate infeasible operation. system. The ideal model presents violations of the technical 9 Fig. 7. Analysis of deviations and unfeasible scheduling for an ideal battery battery power limits, accounted for a 12% deviation from the scheduled battery usage, highlighting the importance of such a detailed model to avoid wrong estimations of attainable flexibility and risking damaging the storage system. Giv en the increasing need for flexibility and reliability in en- ergy systems, the proposed linear model allows characterizing the operation and technical limits of electric battery systems through a computationally-efficient approach. The introduced battery characterization could be applied in combination with online state-of-charge and state-of-health estimators, to mon- itor the battery performance and the capacity fading in the battery . These systems could provide a continuously updating version of the battery capacity and performance, allowing to update the characteristic operation curves for charge and discharge presented in Section IV. 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[26] Gurobi Optimization Inc., “Gurobi Optimization: The state-of-the-art mathematical programming solver, ” http://www .gurobi.com/. [27] A. W ¨ achter and L. T . Biegler, “On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, ” Mathematical Pr ogr amming , vol. 106, no. 1, pp. 25–57, 2006. Alvaro Gonzalez–Castellanos (S’11) received his B.Sc. in electrical engineering from the University of the North, Barranquilla, Colombia, in 2015, the M.Sc. in power engineering from the Skolkovo In- stitute of Science and T echnology , Moscow , Russia, in 2017. Currently , he’ s pursuing a Ph.D. with the Center for Energy Science and T echnology at the Skolko vo Institute of Science and T echnology . His research interests include the con vex characterization of flexibility measures in integrated energy infras- tructures 10 David Pozo (S’06–M’13–SM’18) received his B.S. and Ph.D. degrees in electrical engineering from the University of Castilla-La Mancha, Ciudad Real, Spain, in 2006 and 2013, respecti vely . Since 2017, he is Assistant professor at the Skolkovo Institute of Science and T echnology (Skoltech), Moscow , Russia. Prior to Skoltech Dr . Pozo worked as a post- doctoral fellow at the Pontifical Catholic Univ ersity of Chile and the Pontifical Catholic University of Rio de Janeiro. His research interest lies in the field of power systems and includes operations research, uncertainty , game theory , and electricity markets. He also focuses on problems of optimization and flexibility of modern po wer systems. Since 2018, Prof. Pozo is leading the research group on Power Markets Analytics, Computer Science and Optimization (P ACO). Aldo Bischi , follo wing M.Sc in Mechanical Engi- neering (2007) from University of Perugia (Italy) re- ceiv ed his Ph.D. in Energy and Process Engineering (2012) from the Norwegian University of Science and T echnology-NTNU (Norway). From 2012 until 2015 he held a post-doctoral position at Politecnico di Milano (Italy). Since 2016 he is Assistant Profes- sor at Skolkov o Institute of Science and T echnology (Russian Federation), part-time since 2018 when he started working as a postdoctoral research associate at Uni versity of Pisa (Italy). Dr . Bischi scientific interest lies in the field of energy con version systems, ranging from modeling and experimental activities to optimal scheduling and design, including all kinds of energy vectors such as gas, heat and electricity . 11