CE295 Homework 1- Battery Modeling, Analysis, and Simulation Solution

This assignment will provide hands-on practice for modeling, linearization, stability analysis, and simulation.
Problem 1: Review Submission Procedure from HW0 (ungraded)
Problem 2: Reading
Read Chapter 3 “Drivers for Change” in the following report available on bCourses
• A. N. Annaswamy, M. Amin, C. L. DeMarco, and T. Samad, “IEEE Vision for Smart Grid Controls:
Please summarize in one paragraph or two what recent issues have spurred growth and interest in developing a smart grid. How can modeling, controls, and optimization address these issues?
Problem 3: Black-box vs. White-box Modeling
Please complete the table below in your report, which summarizes the advantages and disadvantages of black-box and white-box modeling. Provide at least three items in each cell.

Black-Box Models White-Box Models
Advantages
Disadvantages

Problem 4: Mathematical Modeling Uses
What are the five potential uses of a mathematical model? List all five uses, and explain.
Problem 5: Mathematical Modeling
Suppose you work at an energy consulting firm contracted by Tesla Motors. Tesla is interested in utilizing their vehicles as a distributed energy storage source that can provide services to the smart grid. This provides their customers with an additional revenue stream to offset their EV investment. To this end, you wish to

Figure 1: OCV-R-RC Equivalent circuit model of a battery. Includes an open circuit voltage in series with a resistance, in series with a resistor-capacitor pair.
develop a mathematical model of a battery that predicts its state-of-charge evolution, for analysis and control purposes.
Batteries can be modeled by a so-called “equivalent circuit model”, shown in Fig. 1. This model idealizes a battery cell as a voltage source OCV in series with a resistor R1 and resistor-capacitor (R2,C) pair. The battery terminal voltage is V (t) and current is I(t). The voltage source OCV is not a constant voltage source. In fact, it is a nonlinear function of the battery state-of-charge (SOC), denoted by z. Moreover, it has the following integrator dynamics
(1)
where Q is the charge capacity of the battery.
(a). List the “stocks” in this model. That is, what are the states?
(b). Use Kirchoff’s voltage and current laws to derive the system equations. Include the resulting equations and integrator dynamics in your report.
(c). List all the parameters, θ, of the model.
(d). Consider current I as the controllable input, and voltage V as the measurable output. Reformulate the model into a state-space form, as discussed in class. You should have multiple state equations and one output equation.
(e). Does the model have a linear state equation? If so, then what are the A,B matrices when written in vector-matrix form? Does the model have a linear output equation? If so, then what are the C,D matrices when written in vector-matrix form? If not, then what term produces the nonlinearity?
Problem 6: Stability and Linearization
In the following, consider the following parameter values: R1 = 0.05Ω,R2 = 0.005Ω,C = 500F,Q = 3600. In addition suppose that the nonlinear function OCV (z) can be expressed by the polynomial
OCV (z) = p0 + p1z + p2z2 + p3z3 (2)
(b). Linearize the output equation from Problem 5(d), using (2) to describe OCV (z), around zeq = 0.5, Vceq = 0, and Ieq = 0. Show all steps for full credit.
Problem 7: Simulation and Analysis
This problem provides practical experience implementing and analyzing your two models. You will be provided skeleton code, HW1_Skeleton.m and HW1_Skeleton.ipynb, on bCourses to assist you.
(a). Enter the missing model parameters into your code, and copy “part(a)” of your code into your report. Also provide a plot of the nonlinear OCV function versus state-of-charge.
(b). Simulate the model with the true nonlinear output function, and the linearized output function. Provide one figure with three subplots: current vs. time, SOC vs. time, and voltage vs. time. On the third subplot, plot both the nonlinear and linearized voltage response. Use legends.
(c). Comment on why the linearized model incurs errors when SOC drops below 25%. Explain using the OCV versus SOC plot. Over time, does the system move away from the linearization point? What are the implications for doing any optimization or control designs with a linearized model?
Deliverables
Submit the following on bCourses. Be sure that all files are named exactly as specified (including spelling and case), and make sure the function declaration is exactly as specified.
LASTNAME_FIRSTNAME_HW1.PDF
LASTNAME_FIRSTNAME_HW1.m or LASTNAME_FIRSTNAME_HW1.ipynb