With renewable energy based electrical systems becoming more prevalent in homes across the globe, microgrids are becoming widespread and could pave the way for future energy distribution. Accurate and economical sizing of stand-alone power system components, including batteries, has been an active area of research, but current control methods do not make them economically feasible. Typically, batteries are treated as a black box that does not account for their internal states in current microgrid simulation and control algorithms. This might lead to under-utilization and over-stacking of batteries. In contrast, detailed physics-based battery models, accounting for internal states, can save a significant amount of energy and cost, utilizing batteries with maximized life and usability. It is important to identify how efficient physics-based models of batteries can be included and addressed in current grid control strategies. In this paper, we present simple examples for microgrids and the direct simulation of the same including physics-based battery models. A representative microgrid example, which integrates stand-alone PV arrays, a Maximum Power Point Tracking (MPPT) controller, batteries, and power electronics, is illustrated. Implementation of the MPPT controller algorithm and physics-based battery model along with other microgrid components as differential algebraic equations is presented. The results of the proposed approach are compared with the conventional control strategies and improvements in performance and speed are Batteries have been integrated in microgrids to mitigate intermittent characteristics of alternative energy sources such as solar, wind, and wave, thereby enhancing grid operation and reliability.1,2 They are well suited for microgrid applications due to their versatility, high energy density, and efficiency. 3 The cost of batteries continues to decrease while their performance and life have continued to increase. 4 However, lithium-ion batteries, which are the most widely used energy storage systems implemented in microgrids today, are still the most expensive component, accounting for about 60% of the overall Capital Expenditure (CapEx).5 Conservative operations in current microgrids cause high cost and low energy efficiency, underutilizing and overstacking batteries. The current microgrid controls cannot utilize batteries aggressively to achieve high penetration of renewables and maximize life and usability of batteries in the meantime. They implement empirical/equivalent-circuit battery models, treating batteries as just a black box, which does not account for its internal states, and place batteries in a small portion of the entire microgrid, which means current microgrids do not consider batteries principal components. [6][7][8][9][10] For example, if the internal temperature of the battery is not modeled, then the battery must be operated at very low rates to ensure that the internal temperature does not reach high enough values that reduce battery life and create unsafe operating c...

A hybrid analytical-collocation approach for fast simulation of the impedance response for a Li-ion battery using the pseudo-two dimensional model is presented. The impedance response of the spherical diffusion equations is solved analytically and collocation is performed on the resulting boundary value problem across the electrode and separator thickness using an orthogonal collocation scheme based on Gauss-Legendre points. The profiles for a frequency range from 0.5 mHz to 10 kHz are compared with the numerical solution obtained by solving the original model in COMSOL Multiphysics. The internal variable profiles across a wide range of frequencies are compared between the two methods and the accuracy, robustness, and computational superiority of the proposed hybrid analytical-collocation approach is presented. The limitations of the proposed approach are also discussed. A freeware for academic use that reads the various battery parameters and frequencies of interest as input, and predicts the battery impedance for a half cell and full cell, is also developed and a means to access it is reported in this paper.

The title compound [systematic name: 2,2′-dinitro-4,4′-(propane-2,2-di­yl)diphenol], C15H14N2O6, crystallizes with two mol­ecules in the asymmetric unit. Both have a trans conformation for their OH groups, and in each, the two aromatic rings are nearly orthogonal, with dihedral angles of 88.30 (3) and 89.62 (2)°. The nitro groups are nearly in the planes of their attached benzene rings, with C—C—N—O torsion angles in the range 1.21 (17)–4.06 (17)°, and they each accept an intra­molecular O—H⋯O hydrogen bond from their adjacent OH groups. One of the OH groups also forms a weak inter­molecular O—H⋯O hydrogen bond.

There are a wide range of battery models at different scales, from empirical models to molecular dynamics models, that can describe the battery behavior. The Pseudo 2-Dimensional (P2D) model considers the porous electrode theory, concentrated solution theory, Ohm’s law, kinetic relationships, as well as charge and material balances.1 These physic-based behaviors are described by a set of stiff nonlinear partial differential algebraic equations (DAEs) which can be only solved numerically. In this talk, we discuss and review different methods to simulate the battery models, specifically about integration in the time domain. As the P2D model discretized in spatial coordinates by using any suitable method such as finite difference2-3, finite volume4 and spectral methods5-8, it results in a system of nonlinear DAEs. Typically, nonlinear DAEs can be solved based on Runge-Kutta (RK) methods (explicit or implicit) or multistep methods. The different orders of accuracy will affect the accuracy of the numerical solutions for each time step and it will further affect the computational efficiency. However, most of the time the stability region becomes smaller when the order of accuracy of the method increases. For the multistep method, Backward Differentiation Formula (BDF) method is commonly used, because we can get more than second order of accuracy without increasing the number of variables.9 Both RK and BDF methods will be reviewed for simulating battery models. Subtle differences among different methods and efficiency improvement and robustness of all these methods will be analyzed and discussed. In particular, the compromise between, stability, accuracy, and ease of programming will be discussed. Implementation in different programming languages and platforms will also be compared. Acknowledgements The authors are thankful for the financial support of this work by the Clean Energy Institute (CEI) at the University of Washington.   References 1. W. Tiedemann and J. Newman, J. Electrochem. Soc., 122, 1482–1485 (1975) 2. M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140, 1526–1533 (1993). 3. G. G. Botte, V. R. Subramanian, and R. E. White, Electrochim. Acta, 45, 2595–2609 (2000). 4. M. Torchio, L. Magni, R. B. Gopaluni, R. D. Braatz, and D. M. Raimondo, J. Electrochem. Soc., 163, A1192–A1205 (2016). 5. V. R. Subramanian, V. Boovaragavan, V. Ramadesigan, and M. Arabandi, J. Electrochem. Soc., 156, A260–A271 (2009). 6. P. W. C. Northrop, V. Ramadesigan, S. De, and V. R. Subramanian, J. Electrochem. Soc., 158, A1461–A1477 (2011). 7. V. Ramadesigan, V. Boovaragavan, J. C. Pirkle, and V. R. Subramanian, J. Electrochem. Soc., 157, A854–A860 (2010). 8. V. R. Subramanian, V. D. Diwakar, and D. Tapriyal, J. Electrochem. Soc., 152, A2002–A2008 (2005). 9. K. Brenan, S. Campbell, and L. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (1995).

The idea of using modeling in battery design was first introduced by W. Tiedemann and J. Newman in 19751. They used an ohmically limited porous electrode model to maximized the cell effective capacity by controlling electrode thickness and porosity. Newman later applied the reaction-zone model to maximize the specific energy of the system by considering mass as well2. V. Ramadesigan et al.3went one step further with the ohmic resistance model in the electrode by including electrode kinetics. They assumed the linear kinetics, and minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using layered graded electrode to get a porosity distribution to further reduce the ohmic resistance in the system. Since the introduction of the pseudo-2D (P2D) model3, many efforts have been made in applying the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method4. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis5 with expanded P2D model incorporating mechanical stress-strain relationship. S. De and his coworkers6 used a reformulated model developed by the Subramanian group7, to perform simultaneous optimization of multiple design parameters. N. Xue et al. 8 applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan9used a gradient-free direct search method to maximize the specific energy. Generally, there are two purposes of using models to simulate the battery system; one is to develop a better understanding of the system, and the other is to use the models as guidance to achieve better performance. For the latter purpose, a model with fewer assumptions and more equations capturing as many processes as possible is desired to simulate the system better. However, for a better understanding of the system, it may be worth taking a step back, and examining a simpler model where the effects of different processes can be observed clearly, even though more detailed physical models with much less assumptions would be used to guide the actual design. This is the purpose and idea behind this study. The model used in this study is an ohmic resistance model, with the assumption that the electrolyte concentration is uniform and double-layer charging can be ignored. Ohm’s law for both solid and electrolyte phases was included in the model, as well as a polarization equation representing the electrode kinetics. The polarization equation describes the charge transfer between the two phases, of which the linear form together with Ohm’s law for the two phases can be found in Ref10. In such a model, the trade-off between ohmic potential drop and reaction kinetics can be captured by current distribution between the two phases. In this work, we want to quantify the impact of nonlinearity on the architecture design by carefully examining the effect of nonlinearity on the ohmic resistance model. For linear kinetics, the optimal design to minimize overall resistance is independent of operation conditions (Fig. 1). However, for nonlinear kinetics, the optimal porosity is bigger for higher charge/discharge rates. This suggests that for batteries built for different applications, the optimal design varies depending on the operation conditions, especially at high rates, where the kinetics fall outside the linear regime. Figure 1 . Optimal Porosity under Different Charge/Discharge Rates for Linear and Nonlinear Polarization Acknowledgements The authors are thankful for the financial support of this work by the Clean Energy Institute (CEI) at the University of Washington. References 1. W. Tiedemann and J. Newman, J. Electrochem. Soc., 122, 1482–1485 (1975) 2. J. Newman, J. Electrochem. Soc., 142, 97 (1995). 3. V. Ramadesigan, R. N. Methekar, F. Latinwo, R. D. Braatz, and V. R. Subramanian, J. Electrochem. Soc., 157, A1328 (2010). 4. W. Du, A. Gupta, X. Zhang, A. M. Sastry, and W. Shyy, Int. J. Heat Mass Transf.,53, 3552–3561 (2010) 5. S. Golmon, K. Maute, and M. L. Dunn, J. Power Sources, 253, 239–250 (2014) 6. S. De, P. W. C. Northrop, V. Ramadesigan, and V. R. Subramanian, J. Power Sources, 227, 161-170 (2013). 7. P. W. C. Northrop, V. Ramadesigan, S. De, and V. R. Subramanian, J. Electrochem. Soc., 158, A1461 (2011). 8. N. Xue et al., J. Electrochem. Soc., 160, A1071–A1078 (2013) 9. Y. Dai and V. Srinivasan, J. Electrochem. Soc., 163, A406–A416 (2015). 10. J. Newman, K. E. Thomas, H. Hafezi, and D. R. Wheeler, J. Power Sources, 119-121, 838–843 (2003). Figure 1