ME 565: Battery Systems and Control, University of Michigan, Winter 2019

Instructor: Jason Siegel (siegeljb), Student Instructor: Ting Cai (tingcai)

HW #4
Due Sunday February 10 for on campus students, and Sunday February 10 for off campus students.
Submission online through Canvas before midnight.

Problem 1: CCCV Problem

The standard battery charging method uses an intuitively-named protocol called “constant current, constant
voltage” (CCCV) to avoid overcharging while maximizing the stored energy in the battery. That is, charge the
battery at a constant current rate, and then lower gradually the current input to maintain a constant voltage
and continue “stuffing” energy into the battery until the battery is fully charged.
Apply a constant charging current of 4.5 A until the voltage reaches the desired voltage of Vdes = 3.6
V whereupon switch to constant voltage mode. That is, design a controller that would automatically adjust
the current, with a battery voltage reference Vdes . Stop charging when the current decays to -30 mA. Since
this indicates that the battery cannot accept any appreciable charge beyond Vdes (anymore charge would raise

m
er as
the battery voltage to higher, unacceptable values). To accomplish this follow these steps (use the model
HW4_virtual_testbed.slx):

co
eH w
(a) In order to build the CCCV controller first start with the HW4_virtual_testbed model and build an error

o.
signal between the desired voltage and actual voltage as e = Vdes − V . Consider the integral controller
rs e
R
I = −KI edt, with initial condition I0 = -4.5 A. Add a saturation block on the current signal going
ou urc
to the battery limit the current above -4.5A. Submit (present a figure) of your completed model that
performs the cc-cv charge with cutoff logic. Place your name in a text box on the top of the integrator.
Note that, in this problem, we are assuming that charging current is negative.
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(b) Simulate a 4.5 A CCCV charge cycle with the integral voltage controller described above. Initialize the
aC s

SOC to 0% (SOC_0=0). Use a control gain KI = 1. Terminate the simulation once the current drops below
vi y re

30 mA in constant voltage mode. Submit plots of time vs. terminal voltage and time vs. applied current
for that gain. Why doesn’t the battery stop charging when the voltage reaches 3.6 V and what happened
to the CV phase? (Hint check the integrator state).
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(c) To address wind-up of the integrator consider an anti-windup scheme. Modify the error signal to stop
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growing once the current saturates. Use the modified error e = Vdes − V + Kaw (I − sat(I)) and simulate
a 4.5 A CCCV charge cycle with the integral voltage controller described above. Initialize the SOC to 0%
(SOC_0=0). Use a control gain KI = 1, and Kaw = 5. Terminate the simulation once the current drops
below 30 mA in constant voltage mode. Submit plots of time vs. terminal voltage and time vs. applied
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current for that gain.

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(d) Try different control gains KI = (0.5, 1, 10, 50, 100) and compare total charge time and voltage overshoot
beyond the maximum allowable voltage. Specifically, submit a table listing controller gains and the
corresponding charge times and voltage overshoots. What can you say about the effect of the different
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gains?

(e) Assuming the OCV-R Equivalent circuit model (ECM) defined by

I
ż = −
Qmax · 3600
V =OCV (z) − I · R

is accurate in modeling the dynamics of the cell voltage vs current, answer the following questions.
Find the denominator of the closed loop transfer function from the voltage set point to the battery voltage
R
for the following two controllers: I = −KP e (proportional control) and I = −KI edt (integral control).
You can ignore the anti-windup and saturation blocks for this question.

https://www.coursehero.com/file/38102623/HW4-2019-V2-1pdf/
 (f) (Bonus) Plot the Root-Locus diagram (rltool) for the closed loop system and discuss how the controller
gain, KI can affect the overshoot, rise time, and stability of the voltage response. consider the slope of
the OCV curve at the SOC when the battery hits 3.6V under a 1C rate to be α = dOCV dz
(z)
= 6, R = 0.01
and Qmax = 4.5.

Problem 2: Impedance of the battery Model

So far, you have identified the OCV-R and the OCV-R-RC models based on their response to signals in time
domain. It is interesting to investigate the model dynamics by looking at the impedance, Z, which is defined by

V (jω) = Z(jω)I(jω), (1)

where ω is the frequency of the signals. The impedance reflects the response of the battery to sinusoidal current.
Compare the impedance of the two models we have developed so far, which is the OCV-R model,

dSOC I
=−
dt Q (2)

m
V = OCV (SOC) − Rs · I

er as
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and the OCV-R-RC model,

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dSOC I
=−

o.
rs e dt Q
dVc Vc I (3)
=− +
ou urc
dt R1 C1 C1
V = OCV (SOC) − Vc − Rs · I.
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Assume that the open circuit voltage OCV(SOC) in both cases is defined as:
aC s

OCV (SOC) = 3.1264 + 3.0532 × SOC − 5.2313 × SOC 2 + 3.2152 × SOC 3 (4)
vi y re

Also assume that the values of the parameters in the above equations, are:

Rs = 0.03Ω, R1 = 0.08Ω, C1 = 5000F, Q = 4.5Ah (5)

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Use the skeleton m-file HW4_Skeleton_Prob2.m to complete this problem.

(a) As the first step of deriving the model impedance, you need to linearize the open circuit voltage around
a certain SOC. Calculate the slope of the OCV curve at 50% SOC based on (4), and provide
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the equations of the linearized OCV-R and OCV-R-RC models. Show the equations with
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the plugged in parameter values.

(b) Derive the expression for the impedance of the OCV-R and the OCV-R-RC models. Separate
the real and the imaginary parts. Show the equations with the plugged in parameter values.
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(c) Show the Nyquist plot of the impedance of the OCV-R model at 50% SOC. The frequency
range is 0.01mHz - 1kHz. Please plot −RE(Z) along the real axis, and IM (Z) along the imaginary axis.
Denote where are f = 0.01mHz and f = 1kHz.

(d) Show the Nyquist plot of the impedance of the OCV-R-RC model at 50% SOC. Requirements
are the same as in (c).

Keep your explanations, your m-file (printed commands), your sml figure and all your plots
all together for each question of each problem and presented in a coherent (sequential flow).
No explanations → no credit. Having answers and plots of all problems in separate area from
explanations and commands or simulink files in your submitted solutions might not be graded.
Please help our graders do their best.

https://www.coursehero.com/file/38102623/HW4-2019-V2-1pdf/

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