Direct, Efficient, and Real-Time Simulation of Physics-Based Battery Models for Stand-Alone PV-Battery Microgrids

PV arrays generate power by converting solar energy directly to electricity. Typically, simulation of equivalent circuit-based models featuring one to three diodes has been the common approach to predict the PV arrays' performance.¹² For mathematical modeling of PV arrays, a single-diode equivalent circuit-based model is used (see Figure 1). This model offers a good balance between simplicity and accuracy by adjusting parameters and modifying the saturation current function to match the open-circuit voltage of the model with experimental data from the solar arrays for a broad range of temperatures. The I-V characteristics of the practical PV device, consisting of multiple cells in a parallel-series combination, is given as:¹³

I is the PV arrays' output current, I_PV is the PV current, I₀ is the saturation current, and V_t is the thermal voltage. These variables are temperature-dependent, but simulation and control are conducted under the assumption that temperature is a constant (see Table III). V is the PV arrays' output voltage, R_s is the series resistance, R_p is the parallel resistance, and a is the diode ideality constant. Meanwhile, the PV arrays' output current and voltage and PV current in Equation 1 can be expressed as a function of time by substituting I = I(t), I_PV = I_PV(t), and V = V(t) as follows:

All additional equations, variables, and parameters are shown in Table I, II, and III, respectively.¹³

Table I. Equations for PV systems.

Governing equation	Additional equation

MPPT controllers can find Maximum Power Points (MPPs), and help PV arrays to produce peak power outputs. Various conventional MPPT algorithms such as P&O and Incremental Conductance (ICond) have widely been implemented as feedback algorithms. The P&O has been one of the most commonly used MPPT algorithms to track MPPs.¹⁴ The P&O algorithm applies iterative processes toward MPPs by adjusting the output voltage of PV arrays with time and voltage step. The PV power at the present instant is obtained by measuring the value of the PV voltage and current, and the power at a future instant is compared to the present power after increasing or decreasing the voltage step. In the case of positive ΔP (difference in values between the future power and present power), the algorithm operates PV systems toward the optimal point by adding the voltage step. Otherwise, if ΔP is negative, the voltage value of the optimal point is determined by subtracting the voltage step. In the P&O algorithm, as the time step increases, simulation time decreases, while accuracy decreases, and vice versa.¹⁵ Also, the range of oscillation has a proportional relationship with the voltage step.¹⁵ The flow chart of the P&O algorithm is presented in Figure 2. Even though the P&O algorithm is not able to track true MPPs, loses some amount of power due to oscillation around the MPP, and fails to track true MPPs because of its slow response to rapidly changing environmental conditions, the algorithm is the most popular, thanks to its simplicity and low hardware cost.^16–20

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In this section we present a DAE-based implementation of MPPT algorithm. MPPs can be calculated from the following formula²¹

in which P(V) is the PV arrays' output power, I(V) is the PV arrays' output current, and V is the PV arrays' output voltage. Note that both P(V) and I(V) are functions of the output voltage. We get Equation 4 by arranging Equation 3 as follows:

Also, the PV arrays' output current in Equation 1 can be expressed as a function of voltage by substituting I = I(V) as follows:

By differentiating Equation 5 with respect to V, we get

By substituting Equation 4 into Equation 6, we get

Finally, substituting I(V) = I(t), V = V(t) in Equation 7 gives

The DAE-based MPPT algorithm consists of two Algebraic Equations (AEs) in two variables (Equations 2 and 8; voltage and current as a function of time, respectively). The system's response time will significantly slow down in a real situation that involves dynamic environmental condition and long time-scale. For instance, one can use nonlinear equations solvers in computational software programs such as Maple, Maplesim, Matlab, and Simulink. However, it would fail or need significantly high computation time due to the solvers' approach in finding the solution.²² Here, we implement the DAE-based MPPT algorithm consisting of AEs only. In this case, one arbitrary Ordinary Differential Equation (ODE) (Equation 9), which does not affect any processes of the entire microgrid system, can be added to convert the AE system into a DAE system as shown in Figure 3.

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An example of a short time simulation with the P&O and DAE-based MPPT algorithm is shown in Figure 4. This will help analyze the DAE-based algorithm's performance including its accuracy, reliability, and computation time. The voltage and time step in the P&O algorithm were adjusted (0.01 V and 0.001 s) to produce high accuracy, thereby explicitly identifying the CPU time difference between the P&O and DAE-based MPPT algorithms. Equations 2, 8, and 9 were implemented for the DAE-based MPPT algorithm. At the beginning of the simulation, when t = 0, the irradiance is assumed to be 400 W/m² and it is increased up to 800 W/m² within a very short time. After that, the irradiance is kept at 800, 400, 1000, 50, 600, and 200 W/m² for 10 seconds each and decreased to 100 W/m² as illustrated in Figure 4a. Even though the whole time scale is small (60 seconds), rapidly changing irradiance from 50 to 1000 W/m² was tested for the P&O and DAE-based MPPT algorithms' reliability. Also, solar output powers are plotted over voltage outputs at each irradiance level from Equation 5 as shown in Figure 4b. In Figures 4c and 4d, voltage and current output responses are presented by implementing the P&O and DAE-based MPPT algorithms. All MPPs in Figure 4b were rounded off to the 6^th decimal place, and all the values of MPPs from the DAE-based MPPT algorithm at each irradiance level exactly matched to the results from Equation 5 with low CPU time (∼0.125 s). On the other hand, the CPU time of the P&O algorithm was significantly slower (27.971 s) than the DAE-based MPPT algorithm with lesser accuracy. This case study clearly illustrates the DAE-based MPPT algorithm's high accuracy, reliability, and computational speed over the traditional P&O MPPT algorithm.

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A bidirectional DC/DC converter is used for PV systems, stepping up/down voltages from its input to output. It regulates output voltages and maintains a constant voltage across DC bus from varying voltages of PV systems under the assumption that there are no energy losses. Here, the voltage value (V_dc) across the DC bus was fixed at 100 V, which was accomplished by increasing or decreasing the duty cycle, which is performed by using Pulse Width Modulation (PWM) of the square-wave pulses to the switches of the DC/DC converter.²⁴ The equations implemented for the bidirectional DC/DC converter are:²⁵

I_dc, PV(t) is the DC bus current, V_{PV, out}(t) is the output voltage, I_{PV, out}(t) is the output current, and D_PV(t) is the duty ratio of the DC/DC converter, from PV systems. A bidirectional DC/AC inverter, which changes DC to AC or AC to DC, is implemented, since most standard appliances run on AC on the consumer side. Here, the conversion efficiency (η) is fixed as 0.85.

P_out, PV(t) is the output power from the DC/AC inverter, and P_in, PV(t)( = V_dc · I_dc, PV(t)) is the input power from the DC/DC converter connected to PV systems.

The MPPT simulation for PV arrays was conducted for a single day in LA, in April 2010 as shown in Figure 6. PV arrays were scaled up to kilowatt-scale systems. Figure 6a shows that the sun rises at 6 AM, and solar irradiance rises and falls until 7 PM. Under the rapidly changing environmental conditions, voltage and time step in the P&O algorithm need to be calibrated to produce the best performance as shown in Table IV. When the voltage step is 0.1 V, the maximum power output is generated, and computation time decreases with a small decrease in power as the time step increases. In other words, when the time step is increasing from 5 to 20 seconds, power outputs are the maximum in the voltage step of 0.1 V. Also, the time step of 15 seconds has a good balance between computation time and power outputs. Accordingly, the time and voltage step are fixed as 15 seconds and 0.1 V, respectively. For the DAE-based MPPT algorithm, Equations 2, 8, 9, 10, 11, and 12 were implemented. The DAE-based MPPT simulation's total CPU time is 0.702 seconds, which is 12 times faster than the P&O algorithm with more power (P&O: 1307.96 kWh, DAE: 1308.89 kWh), as shown in Figure 6b. Also, while the P&O produces oscillations around MPPs (see Figures 6c and 6d), the DAE-based MPPT algorithm precisely follows MPPs. It is evident that the amount of energy loss will increase as the PV systems' scale increases.

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Implementation of highly non-linearized DAEs in detailed battery models with MPPT feedback algorithms such as P&O and ICond causes a significant time-delay as the controls consume computation time at every time step.^28–30 However, once the DAE-based MPPT algorithm is efficiently simulated as illustrated in the earlier sections, adding efficient models for batteries is straightforward. The mathematical structure of the DAE-based MPPT helps easily add battery and power electronic models in the microgrid architecture (see Equations 2 and 8). For example, even the reformulated P2D model, one of the highly non-linearized DAE battery systems, which includes several important internal variables simulated at a reasonable computation time,²⁶ can be simply incorporated into the DAE-based MPPT algorithm, power electronics, and PV arrays mathematical models. Instead of adding one arbitrary equation (Equation 9), the combination of reformulated P2D battery, power electronics, PV arrays, and the DAE-based MPPT mathematical models, which includes several ODEs and AEs, becomes a system of DAEs as shown in Figure 8. The DAE solvers in the computational software allow the entire microgrid system to be simulated simultaneously with high accuracy and low computation time under rapidly changing environmental conditions.^22,31 As more detailed physics-based battery models are involved in the entire microgrid, the DAE-based MPPT algorithm will be better suited for simultaneous simulations under real environmental conditions.

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An ideal battery model would provide good prediction while maintaining minimal computational cost. The porous electrode P2D model, which is one of electrochemical engineering models, includes several significant internal variables and has a predictive capability for battery simulations.^32,33 The P2D model is one of the most accurate and experimentally validated physics-based battery models that allows for modeling of critical internal states used in the literature. The simulations are conducted on a mathematically reformulated P2D model that enables entire microgrid simulations to be solved in a reasonable time.^26,33 High accuracy and short computation time of the reformulated P2D model are well suited for grid applications.³⁴ The reformulated model discretizes dependent variables as a series of trial functions, rather than a finite difference approach, using coordinate transformation and orthogonal collocation.²⁶ Also, the Solid Electrolyte Interphase (SEI) layer is modeled by considering irreversible side reactions. During charging operations, some of the cyclable lithium ions are removed by reactions between the lithium ions and electrolyte, and the reactions create a passivation layer around the anode, which is described by Butler-Volmer kinetics (see Table VI).^35,36 Equations, variables, and parameters related to the reformulated P2D model are arranged in Tables Vt6t7–VIII.^{26,32,33,35,36}

Table V. Governing equations for Li-ion P2D model.

Governing equation	Boundary conditions
Positive
Separator
Negative electrode

Table VI. Additional equations for Li-ion P2D model.

i = p, s, n

Deff, i = D · εbruggi,  i = p, s, n

σeff, i = σi  ·  (1 − εi − εf, i),  i = p, n

Table VII. List of variables for Li-ion P2D model.

Symbol	Variables	Units
c	Electrolyte concentration	mol/m3
csi	Solid phase concentration	mol/m3
Φ1	Solid phase potential	V
Φ2	Liquid phase potential	V
cs, surf	Solid phase concentration at surface	mol/m3
cs, ave	Average solid phase concentration	mol/m3
I	Applied current density	A/m2
Ui	Open circuit potential at positive (i = p) and negative (i = n)	V
ji	Pore wall flux at positive (i = p) and negative (i = n)	mol/m2/s
κeff, i	Liquid phase conductivity at positive (i = p), separator (i = s), and negative (i = n)	S/m
Deff, i	Effective diffusion coefficient conductivity at positive (i = p), separator (i = s), and negative (i = n)	m2/s
σeff, i	Effective solid phase conductivity at positive (i = p) and negative (i = n)	S/m
θi	State of charge at positive (i = p) and negative (i = n)	-
jSEI	Flux associated with SEI layer growth	mol/m2/s
csol	Concentration of solvent at anode surface	mol/m3
	Concentration of electrolyte at anode surface	mol/m3
δ	SEI layer thickness	m

Batteries are charged through the power from the PV arrays converted by the DC/AC inverter and discharged based on users' demands on the microgrid. The solar energy converted by the DC/AC inverter will be provided directly to users on the microgrid for any demand below the level of the solar power, and batteries will be charged with any power available above the level of demand. In contrast, when the converted solar power decreases below the desired level of demand, batteries will be discharged to provide the stored energy to the users. The bidirectional DC/DC converter is connected to batteries in the same manner as the PV systems' configuration. Also, the role of power electronics connected to batteries, the voltage value across the DC bus, and the bidirectional DC/AC inverter's conversion efficiency are assumed to be the same as the PV systems were described in the previous section. The DC/DC converter's equations are:²⁵

I_{dc, battery}(t) is the DC bus current, V_{battery, out}(t) is the output voltage, I_{battery, out}(t) is the output current, and D_battery(t) is the duty ratio of the DC/DC converter, from batteries. For the bidirectional DC/AC inverter, the amount of DC/AC inverter's output power (P_{out, battery}(t)) supplied by batteries' power (P_{in, battery}(t)) can be calculated based on the relation between the input and output power in Equation 12 as shown below:

P_{out, battery}(t) is the output power from the DC/AC inverter, and P_{in, battery}(t) is the input power from the DC/DC converter connected to batteries. Batteries are operated to make up the difference between the demand (P_demand(t)) and the PV arrays' power (P_{out, PV}(t)) converted by the DC/AC inverter.

By substituting Equations 12 and 16 into Equation 17, we get

The reformulated P2D model for single cathode/separator/anode cell sandwich was scaled up to a kilowatt-scale systems corresponding to the power level of PV systems and demands (assuming no loss in performance). A 1500 kWh sized battery with 10% initial state of charge (SOC) was used in the simulation to illustrate the efficiency of the P2D model along with the fast computation time in proposed microgrid controls. Cost and energy savings by implementing the proposed control strategy including the electrochemical engineering model and DAE-based MPPT algorithm is also presented. The demand starts at 12 PM and continues until 7 PM at a constant value (120 kW), as shown in Figure 9a. The total demand is 840 kWh. The power obtained from solar arrays, using the P&O and DAE-based MPPT algorithm, are 1307.96 kWh and 1308.89 kWh, respectively. The battery will be discharged when the demand is greater than the power supplied from solar arrays, and vice versa (see Figure 9b). In other words, when batteries' current is below zero, which means demand is greater than solar array's power, batteries are discharged. In Figures 9c and 9d, the reformulated P2D model predicts voltage and SOC accurately with reasonable CPU time. The proposed control scheme's running time is only 19.157 seconds for a single day simulation. As we mentioned before, lithium-ion batteries are the most expensive single component accounting for about 60% of overall CapEx among the whole microgrid components.⁵ Total costs of microgrids currently range between 583 and 1166 $/kWh.⁵ Accordingly, total costs of the lithium-ion batteries in CapEx is between 350 and 700 $/kWh. For example, if we can reduce the battery size by 50%, total cost of the battery system will decrease to 175–350 $/kWh with significant reduction in total cost for large-scale microgrids.

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Simple empirical/equivalent-circuit or black-box battery models will not be able to capture the essential dynamics of battery performance accurately and will provide poor estimates of cell size and costs. The proposed approach is fundamentally different from the existing microgrid control architecture. Instead of treating cells as a black box, critical internal states will be addressed with high accuracy and capable of being simulated in real-time. For example, SEI layer growth from the reformulated P2D battery model is presented in Figure 10. The SEI layer causes lithium ions' diffusion resistance and a voltage drop across the battery. Including the SEI layer growth as an internal variable can be used to control the capacity fade over cycles.³⁴ Even though capacity fade/degradation is accelerated by extreme charging patterns, increased temperature, and overcharging, the batteries degrade even under normal operation.

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