Gelöste Aufgaben/JUMP/Battery

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Scope

Our Block-Diagram

The battery is our power-supply. Depending on the discharging current I_B consumed by our drive-train, the battery will display a voltage U_B.

We are looking for a mathematical model that accounts for discharging operations only and we are not interested in high-frequency load-idle-cycles. Long-term changes of battery-parameters are out of scope.

We copy the typical specifications for a LiPo battery used for RC Model Cars to be:

• total capacity: Q_B0 = 5000 mAh,
• voltage: U_B0 = 11.1 V,
• C-rating: 25.

The C-rating refers to the rate of energy the battery can safely discharge continuously, represented in terms of current as a multiple of its overall capacity. The maximum continuous current for this battery is

$\begin{array}{ll}{I}_{B,max}& =5000mAh\cdot 25\cdot 1/h\\ & =125A\end{array}$

Also we need to model the battery heat to avoid excessive temperatures T_B of the battery.

Structure

Our model accounts for two aspects:

1. power delivery and
2. temperature.

Both aspects are related in the sense that discharging operations with high currents lead to high battery-temperatures which will reduce the effective battery capacity.

The equivalent block-diagram of our model is this:

The battery-current I_B entails a battery voltage U_B and Temperature T_B.

Power Delivery

Battery Equivalent Circuit Diagram

We employ a standard battery model which accounts for

• E: the equilibrium potential
• R₁, R₂: two internal resistances and
• C: an effective capacitance.

In a steady-state-condition, the battery voltage U_B is thus a function of the battery current I_B.

Temperature

Battery Heat Rates.

Our simple lumped-body model accounts for heat generation inside the battery and heat dissipation over its surface.

In steady-state condition, they are equal, if heat generation ${\dot{Q}}_H$ exceeds heat dissipation ${\dot{Q}}_D$, the battery heat increases - und thus its temperature.

Model

Power Delivery

Using Kirchhoff’s laws and from the model-diagram of the electric-circuit of the battery, we find the equilibrium conditions

$\begin{array}{ll}{U}_B& =U_1+U_2+E(S_D)\text{ where }E(S_D):=U_E\text{ and }\\ I_B& =I_C+I_{R2}\end{array}$.

The potential E is a function of the effective battery capacity or the commonly used dimensionless “state of dischange”

$S_D=\frac{Q_{B0}-Q_B(t)}{Q_{B0}}$.

Characteristic of Electrochemical Potential.

E(S_D) depends in a complex way on the electrochemical processes within the battery. We account for this functional relationship using a characteristic curve which we copy from literature.

In operation, the reference battery capacity Q_B0 is being consumed by the current I_B, thus

${\dot{Q}}_B=-\alpha (I_B)\cdot \beta (T_B)\cdot I_B$,

where α is a correction-factor accounting for the current I_B and β accounting for the battery-temperature T_B. The effect of α can be seen in the diagram above where I_e > I_ref.

For a representation of this function, we choose an appropriate polynomial with

$\begin{array}{lcl}e(S_D)& :=& \frac{E(Q_B)}{U_0}\\ & =& \sum _0^5{C}_i\cdot S_D^i\end{array}$

Its unknown coefficients C_i will be identified from the characteristic above. The components of the electric circuit provide these equations

$\begin{array}{ll}{U}_1& =R_1\cdot I_B,\\ U_2& =R_2\cdot I_{R2},\\ I_C& =C\cdot \frac{d{U}_2}{dt}.\end{array}$

with α, β being current- and temperature-related correction factors respectively.

With the nominal battery capacity Q_B0 and the actual capacity Q_B, we have the commonly employed state of discharge S_D to be

$\begin{array}{ll}{S}_D& =\frac{1}{Q_{B0}}\cdot \underset{0}{\overset{t}{\int }}\alpha (I_B)\cdot \beta (T)\cdot I_{B}dt\\ & =1-\frac{Q_B(t)}{Q_{B0}}.\end{array}$

/*******************************************************/
/* MAXIMA script                                       */
/* version: wxMaxima 18.10.1                           */
/* author: Andreas Baumgart                            */
/* last updated: 2021-02-08                            */
/* ref: Modelling and Simulation (TUAS)                */
/* description: Rounding between straight line-segments*/
/*******************************************************/

/*******************************************************/
/* declarations                                        */
/*******************************************************/
declare("ω",alphabetic);
declare("α",alphabetic);
declare("β",alphabetic);

/*******************************************************/
/* equations of motion                                 */
/*******************************************************/
/* Kirchhoff's law     */
KL : [U[0] = E+U[1]+U[2],
      U[1] = R[1]*I[1],
      U[2] = R[2]*I[2],
      I[1] = I[2]+I[C],
      I[C] = C*'diff(U[2],t),
      'diff(Q[B],t) = α*β*I[1],
      s = Q[B]/Q[R],
      /*             */
      m[B]*c[PB]*'diff(T[B],t)=R[1]*I[1]^2+R[2]*I[2]^2-H[B]*(T[B]-T[0])];

battery: ratsimp([subst(KL[2],KL[1]),
                  subst(solve(KL[3],I[2]), subst(solve(KL[4],I[C]), solve(KL[5],'diff(U[2],t))[1])),
		  KL[6],KL[7],KL[8]]);

/* Characteristic curve for E */
/*                            */
E(s) := sum(c[i]*s^i,i,0,5);

bc : [subst([s=0],E(s))=V[0],E(1)=0,
      subst([s=0],diff(E(s),s))=S[0],subst([s=1],diff(E(s),s))=S[2],
      E(6/10)=V[1],subst([s=6/10],diff(E(s),s))=S[1]];

params: [V[0]=4,S[0]=-1.5,V[1]=3.7,S[1]=-0.4,S[2]=-30];

sol : solve(subst(params,bc),makelist(c[i],i,0,5))[1];
plot2d(subst(sol,E(s)),[s,0,1], [xlabel,"SoD"], [ylabel,"U[0]"]);

Temperature

The thermodynamic balance-equation of heat yields

${\dot{Q}}_T={\dot{Q}}_H-{\dot{Q}}_D$.

✔ Nomenclature "Q":

Note that Q is used here as the heat of the battery – not as the battery-capacity as above!

The battery heat is a function of

• temperature T_B,
• averaged specific heat capacity c_p,B and
• battery mass m_B.

so that

${\dot{Q}}_T=m_B\cdot c_B\cdot \frac{dT}{dt}$.

Heat generation inside the battery be

${\dot{Q}}_H=R_1\cdot I_B^2+R_2\cdot I_{R2}^2$.

and heat dissipation

${\dot{Q}}_D=h_B\cdot (T_B-T_0)$

with ambient air temperature T₀ and the coefficient of heat transfer h_B.

/* none given */

Summary

Thus, the battery-coordinates are

${\underset{\_}{q}}_B=\left(\begin{array}{c}{Q}_B(t)\\ U_2(t)\\ T_B(t)\end{array}\right)$

with the associated equations of motion

$\begin{array}{lll}{\dot{Q}}_B& =-\alpha (I_B)\cdot \beta (T)\cdot I_B\text{ and }{S}_D& =\frac{Q_B(t)-Q_{B0}}{Q_{B0}},\\ {\dot{U}}_2& =\frac{I_C}{C},\\ {\dot{T}}_B& =\frac{1}{m_B{c}_B}\cdot ({\dot{Q}}_H-{\dot{Q}}_D)\\ \end{array}$

and additional algebraic equation

$\begin{array}{ll}{\dot{Q}}_H& =R_1\cdot I_B^2+R_2\cdot I_2^2\text{ where }{I}_2=\frac{U_2}{R_2}\\ U_B& =R_1\cdot I_B+U_2+E(S_D),\\ {\dot{Q}}_D& =H_B\cdot (T_B(t)-T_0).\end{array}$

Variables

name	symbol	unit
battery remaining capacity	QB	A h
capacitance voltage	UB2	V
battery temperature	TB	°C

Parameters

The coefficients C_i for e(S_D) have been computed from conditions for a 5th-order polynomial by defining function-values and gradients at reference points:

$C_0=\frac{11}{10},C_1=-1,C_2=\frac{1177}{240},C_3=-\frac{28489}{2160},C_4=\frac{7891}{432},C_5=-\frac{4355}{432}$

Implemented Characteristic

The plot right shows the resultant characteristic of e(S_D):

For the remaining parameters, no measurements were available. We adopted those to obtain a performance similar to what has been described in [1], the resistors were chosen as to “deliver” reasonable amounts of heat when the battery is discharged with rating of C=25.

• R₁ = 500 mΩ
• R₂ = 250 mΩ
• C = 500 F
• Q_B0 = 5000 Ah
• T₀ = 20°C

For α and β we know that their gradients

$\frac{d\alpha }{d{I}_B}>0$

and

$\frac{d\beta }{dT}<0$.

The reference curve from above shall be assumed to be valid for I_B = 20 A and we assume

$\begin{array}{lll}\alpha (I_B)& =1+\frac{I_B}{I_{\alpha }}& \text{ with }{I}_{\alpha }=100A\\ \beta (T)& =1-\frac{T_B}{T_{\beta }}& \text{ with }{T}_{\beta }=100^{\circ }C\end{array}$

Then, for the thermal model we have

• m_B = 340 g,
• c_p,B = 2 J/(kg K).

And finally h_B will be identified from a (virtual) experiment: let I_B = 0, I₂=0 (no heat generation) and the initial battery-temperature to be ΔT(0) above T₀. Then assume that ΔT(3min) is reduced to 50% of ΔT(0). From above, we find the differential equation to be

$\mathrm{\Delta }T(t)=\mathrm{\Delta }{T}_0\cdot {\text{e}}^{-\frac{h_B\cdot t}{m_B\cdot c_{PB}}}$,

thus for this “experiment”

$h_B=(\text{log}(2)\cdot m_B\cdot c_{PB})/3\text{ min}$.

With the values given above, we find

h_B=2.6E-3 J/(Ks)

Let the tolerable, maximum battery temperature be

$T_{B,max}=60^{\circ }C$.

name	symbol	value	unit
battery mass	mB	340	g
specific heat capacity (averaged)	cpB	1015	J/(kg*K)
resistor 1	R1	0.05	Ω
resistor 2	R2	0.025	Ω
capacity	QB0	5000	Ah
battery discharge rating	1C	25	A
battery reference potential voltage	U0	8	V
ambient temperature	T0	20	°C
time it takes to let the battery cool down by 50%	T50	180	s
correction value for current IB	Iα	100	A
correction value for current TB	Tβ	100	°C

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References

1. Lijun Gao, Shengyi Liu; Roger A. Dougal: Dynamic Lithium-Ion Battery Model for System Simulation, IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 25, NO. 3, SEPTEMBER 2002 pp. 495
2. P. Villano, M. Carewska, S. Passerini: Specific heat capacity of lithium polymer battery components; Elsevier, Thermochimica Acta 402 (2003) 219–224