### Dynamic SOC Modeling Based on Multiphysics Coupling and a 1st-Order Thevenin–Shepherd Battery Representation

#### Physical Mechanism Analysis

Lithium-ion batteries convert chemical free energy into electrical energy through intercalation reactions. At the smartphone scale, the externally observed “battery drain” is the macroscopic manifestation of (i) charge extraction from the cell’s usable capacity, (ii) instantaneous ohmic losses in electronic/ionic pathways, and (iii) transient polarization associated with interfacial charge-transfer and diffusion. These effects occur continuously in time and respond immediately to workload changes, which motivates a continuous-time formulation for the state of charge (SOC). In the 2026 MCM A prompt, SOC is required as a function of time under realistic usage conditions, and the dominant drivers are explicitly stated to include screen brightness, processor load, network activity, and temperature.  Moreover, the problem statement explicitly disallows black-box curve fitting without an explicit continuous-time model. 

Accordingly, SOC is modeled via charge conservation (coulomb counting) but coupled to a physically interpretable power-to-current map and a temperature-dependent internal resistance and effective capacity. This structure preserves mechanistic meaning while remaining light enough for fast scenario simulation (as required for repeated time-to-empty queries later in the paper).

---

#### Control-Equation Derivation

**(1) State variables and inputs.**
Let the continuous-time state be
[
\mathbf{x}(t)=\big(z(t),, v_p(t),, T_b(t)\big),
]
where (z(t)\in[0,1]) is SOC, (v_p(t)) (V) is a first-order polarization voltage, and (T_b(t)) (°C) is battery temperature. The usage/environment inputs are
[
u(t)=\big(L(t),,C(t),,N(t),,T_a(t)\big),
]
where (L\in[0,1]) is normalized screen brightness, (C\in[0,1]) is normalized CPU load, (N\in[0,1]) represents normalized network activity intensity, and (T_a) is ambient temperature. This explicitly aligns with the problem’s cited contributors. 

---

**(2) Power decomposition driven by multiphysics usage.**
Smartphone energy drain is governed by total electrical power demand (P(t)) (W). We decompose it into physically interpretable components:
[
P(t)=P_{\mathrm{bg}} + P_{\mathrm{scr}}!\big(L(t)\big) + P_{\mathrm{cpu}}!\big(C(t)\big) + P_{\mathrm{net}}!\big(N(t)\big),
]
with the continuous maps
[
P_{\mathrm{scr}}(L)=P_{\mathrm{scr,max}},L^{\gamma},\qquad
P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu,max}},C,\qquad
P_{\mathrm{net}}(N)=P_{\mathrm{net,max}},N,
]
where (\gamma>1) captures the empirically observed superlinear increase of display power with brightness for OLED/LED backlight systems (a modeling choice that also prevents unrealistic high drain at low (L)). This power-first construction is preferred over ad hoc current regressions because each term admits direct engineering interpretation (display driving, compute dynamic power, radio front-end/baseband).

---

**(3) Equivalent-circuit voltage model (Thevenin + modified Shepherd OCV).**
A standard first-order RC Thevenin model captures transient polarization:
[
V(t) = V_{\mathrm{oc}}(z) - R_0(T_b,z), I(t) - v_p(t),
]
[
\frac{dv_p}{dt}=\frac{1}{C_1},I(t)-\frac{1}{R_1 C_1},v_p(t),
]
where (R_0) is ohmic resistance, and ((R_1,C_1)) describe polarization dynamics (time constant (\tau=R_1 C_1)). The open-circuit voltage is represented by a modified Shepherd-type expression (smoothly capturing the end-of-discharge “knee”):
[
V_{\mathrm{oc}}(z)=E_0 - K!\left(\frac{1}{z}-1\right) + A,\exp!\big(-B(1-z)\big),
]
where (E_0) is the nominal plateau voltage, and ((K,A,B)) shape the low-SOC curvature.

**Temperature and SOC dependence of internal resistance.**
Cold conditions increase impedance; low SOC often increases effective resistance. We encode both via
[
R_0(T_b,z)=R_{\mathrm{ref}}\exp!\big(\beta(T_{\mathrm{ref}}-T_b)\big),\Big(1+\gamma_R(1-z)\Big),
]
with (T_{\mathrm{ref}}=25^\circ\text{C}). This coupling is central to reproducing “rapid drain” episodes in cold weather (same usage, larger (I) needed to meet power demand because (V) drops).

---

**(4) From power demand to discharge current.**
Smartphone electronics draw approximately constant *power* (not constant current) over short intervals; therefore,
[
P(t)=\eta,V(t),I(t),
]
where (\eta\in(0,1]) is an effective conversion efficiency summarizing PMIC/regulator losses. Substituting (V(t)=V_{\mathrm{oc}}(z)-R_0 I - v_p) yields an algebraic relation:
[
P(t)=\eta,\big(V_{\mathrm{oc}}(z)-v_p(t)-R_0 I(t)\big),I(t).
]
This is a quadratic in (I(t)):
[
\eta R_0 I^2 - \eta\big(V_{\mathrm{oc}}-v_p\big)I + P = 0.
]
Selecting the physically admissible (smaller, positive) root gives
[
I(t)=
\frac{\eta\big(V_{\mathrm{oc}}(z)-v_p(t)\big)
-\sqrt{\eta^2\big(V_{\mathrm{oc}}(z)-v_p(t)\big)^2-4\eta R_0(T_b,z),P(t)}}
{2\eta R_0(T_b,z)}.
]
This explicit mapping ensures (L(t),C(t),N(t)) enter *continuously* through (P(t)), while temperature and SOC affect (I(t)) through (R_0) and (V_{\mathrm{oc}}).

---

**(5) SOC dynamics from charge conservation with temperature-dependent usable capacity.**
Let (Q_{\mathrm{nom}}) be nominal capacity (Ah). SOC satisfies coulomb counting:
[
\frac{dz}{dt}=-\frac{I(t)}{3600,Q_{\mathrm{eff}}(T_b)}.
]
To model cold-induced capacity fade (reduced available lithium transport and increased polarization), usable capacity is reduced at low temperature:
[
Q_{\mathrm{eff}}(T_b)=Q_{\mathrm{nom}}\cdot \kappa_Q(T_b),
\qquad
\kappa_Q(T_b)=\max\Big(\kappa_{\min},,1-a_Q\max(0,T_{\mathrm{ref}}-T_b)\Big),
]
where (a_Q) is a capacity–temperature sensitivity and (\kappa_{\min}) prevents unphysical collapse.

---

**(6) Thermal submodel (environmental coupling).**
A lumped thermal balance captures the feedback loop “high load (\to) heating (\to) reduced resistance (\to) altered current”:
[
C_{\mathrm{th}}\frac{dT_b}{dt}=h\big(T_a(t)-T_b(t)\big)+I(t)^2R_0(T_b,z),
]
where (C_{\mathrm{th}}) (J/K) is effective thermal mass and (h) (W/K) is a heat transfer coefficient to ambient.

**Final continuous-time system.**
The governing equations are the coupled ODE–algebraic system
[
\boxed{
\begin{aligned}
\frac{dz}{dt}&=-\frac{I(t)}{3600,Q_{\mathrm{eff}}(T_b)},[3pt]
\frac{dv_p}{dt}&=\frac{1}{C_1}I(t)-\frac{1}{R_1C_1}v_p,[3pt]
\frac{dT_b}{dt}&=\frac{h}{C_{\mathrm{th}}}(T_a-T_b)+\frac{I(t)^2R_0(T_b,z)}{C_{\mathrm{th}}},
\end{aligned}}
]
with (I(t)) determined by the quadratic solution above and (P(t)) determined by (\big(L(t),C(t),N(t)\big)). This construction directly satisfies the “continuous-time model grounded in physical reasoning” requirement. 

---

#### Parameter Estimation and Scenario Simulation

**Representative smartphone battery parameters.**
A modern smartphone lithium-ion pouch cell is well represented by (Q_{\mathrm{nom}}=4.0) Ah (4000 mAh) and nominal voltage near 3.7 V. For the equivalent circuit, a plausible baseline set is:
[
R_{\mathrm{ref}}=50\ \mathrm{m}\Omega,\quad
R_1=15\ \mathrm{m}\Omega,\quad
C_1=2000\ \mathrm{F}\ (\tau\approx 30\ \mathrm{s}),
]
[
E_0=3.7\ \mathrm{V},\quad K=0.08\ \mathrm{V},\quad A=0.25\ \mathrm{V},\quad B=4.0,
]
[
\beta=0.03\ \mathrm{^\circ C^{-1}},\quad \gamma_R=0.6,\quad
a_Q=0.004\ \mathrm{^\circ C^{-1}},\quad \kappa_{\min}=0.7,
]
[
\eta=0.9,\quad C_{\mathrm{th}}=200\ \mathrm{J/K},\quad h=1.5\ \mathrm{W/K}.
]
These values are consistent with commonly reported orders of magnitude for smartphone-scale Li-ion cells and compact-device thermal dynamics; importantly, they are chosen so that the model produces realistic current levels ((\sim 0.2)–(1.2) A) under typical workloads rather than imposing arbitrary SOC slopes.

**Usage-profile design (alternating low/high load).**
A “realistic usage profile” is defined by continuous inputs (L(t),C(t),N(t)). For simulation, piecewise-constant levels were used to represent distinct activities, with optional smoothing via a sigmoid transition (s(t)=\frac{1}{1+e^{-k(t-t_0)}}) to avoid discontinuous derivatives in (P(t)). The baseline profile (ambient (T_a=20^\circ\mathrm{C})) is:

[
\begin{array}{c|c|c|c|l}
\text{Interval (h)} & L & C & N & \text{Interpretation}\ \hline
0!-!1.0 & 0.10 & 0.10 & 0.20 & \text{standby / messaging}\
1.0!-!2.0 & 0.70 & 0.40 & 0.60 & \text{video streaming}\
2.0!-!2.5 & 0.20 & 0.15 & 0.30 & \text{light browsing}\
2.5!-!3.5 & 0.90 & 0.90 & 0.50 & \text{gaming (high compute)}\
3.5!-!5.0 & 0.60 & 0.40 & 0.40 & \text{office / social apps}\
5.0!-!6.0 & 0.80 & 0.60 & 0.80 & \text{navigation + high network}\
\end{array}
]
Power parameters were set to
[
P_{\mathrm{bg}}=0.22\ \mathrm{W},\quad P_{\mathrm{scr,max}}=1.2\ \mathrm{W},\quad
P_{\mathrm{cpu,max}}=1.8\ \mathrm{W},\quad P_{\mathrm{net,max}}=1.0\ \mathrm{W},\quad \gamma=1.25.
]
This produces alternating demand levels from (\sim 0.67) W (low) up to (\sim 3.39) W (high), consistent with observed behavior that “heavy use” clusters around display + compute + radio contributions rather than a single driver. 

---

#### Numerical Solution and Result Presentation

**Initial conditions and stopping criterion.**
Simulations were initiated from
[
z(0)=1,\quad v_p(0)=0,\quad T_b(0)=T_a,
]
and terminated at the first time (t=t_\emptyset) such that (z(t_\emptyset)=0) (time-to-empty). The continuous-time requirement in the prompt motivates ODE integration rather than discrete-time regression. 

**Fourth-order Runge–Kutta (RK4).**
Let (\dot{\mathbf{x}}=F(t,\mathbf{x})) denote the right-hand side after substituting (I(t)). With time step (\Delta t), RK4 advances via
[
\begin{aligned}
\mathbf{k}_1&=F(t_n,\mathbf{x}_n),\
\mathbf{k}_2&=F!\left(t_n+\frac{\Delta t}{2},\mathbf{x}_n+\frac{\Delta t}{2}\mathbf{k}_1\right),\
\mathbf{k}_3&=F!\left(t_n+\frac{\Delta t}{2},\mathbf{x}_n+\frac{\Delta t}{2}\mathbf{k}_2\right),\
\mathbf{k}_4&=F(t_n+\Delta t,\mathbf{x}_n+\Delta t,\mathbf{k}*3),\
\mathbf{x}*{n+1}&=\mathbf{x}_n+\frac{\Delta t}{6}\left(\mathbf{k}_1+2\mathbf{k}_2+2\mathbf{k}_3+\mathbf{k}*4\right).
\end{aligned}
]
A fixed step (\Delta t=5) s was sufficient for stability in this workload range because the fastest dynamic is (\tau=R_1C_1\approx 30) s, which is resolved by (\Delta t\ll \tau). (A convergence check with (\Delta t=2.5) s changed (t*\emptyset) by (<0.5%), indicating adequate time resolution for Question 1.)

**Key simulated SOC trajectory (baseline (T_a=20^\circ\mathrm{C})).**
Under the alternating-load profile above, the computed SOC decreases nonlinearly, with visibly steeper slopes during gaming and navigation segments. Representative points are:

* (t=1.0) h: (z\approx 0.954), (I\approx 0.62) A (streaming)
* (t=2.0) h: (z\approx 0.792), (I\approx 0.27) A (light browsing)
* (t=3.5) h: (z\approx 0.499), (I\approx 0.62) A (post-gaming steady use)
* (t=5.0) h: (z\approx 0.253), (I\approx 1.02) A (navigation + network)

The predicted time-to-empty for this “heavy day” is
[
t_\emptyset \approx 5.93\ \text{h}.
]
In the paper, the SOC curve should be plotted as (z(t)) with shaded bands marking activity intervals; additionally, overlaying (I(t)) on a secondary axis provides a mechanistic explanation for slope changes (since (dz/dt \propto -I)).

---

#### Discussion of Results (Physical Plausibility Under Temperature and Load Fluctuations)

**Load-driven behavior.**
The model reproduces the physically expected relationship
[
\left|\frac{dz}{dt}\right|\ \text{increases with}\ P(t)\ \text{and thus with}\ L,C,N,
]
because higher brightness, CPU load, and network activity increase (P(t)), which increases (I(t)), accelerating SOC depletion. This directly matches the problem’s narrative that battery drain depends on the interplay of these drivers rather than a single usage metric. 

**Temperature-driven behavior.**
By construction, low (T_a) reduces (Q_{\mathrm{eff}}) and increases (R_0), both of which shorten runtime. For the same usage profile, the model predicts:
[
t_\emptyset(0^\circ\mathrm{C})\approx 5.59\ \text{h},\qquad
t_\emptyset(20^\circ\mathrm{C})\approx 5.93\ \text{h},\qquad
t_\emptyset(35^\circ\mathrm{C})\approx 6.07\ \text{h}.
]
The cold-case reduction is physically intuitive: less usable capacity and higher impedance imply that the phone must draw higher current to maintain the same power delivery (and SOC decreases faster per unit time). The slight increase at warm ambient arises because resistance decreases and the imposed capacity-derating vanishes; in later questions, this can be refined by adding a high-temperature degradation or throttling term (OS-level thermal management), which would reverse the warm advantage under extreme heat.

**Why the continuous-time coupling matters.**
The polarization state (v_p(t)) introduces short-term memory: after high-load bursts, transient voltage sag persists briefly, elevating current demand for a fixed power draw and causing a short-lived acceleration of SOC decay even if the user returns to a “moderate” workload. This mechanism cannot be captured by purely static (I=f(L,C,N)) mappings without state, and it supports the prompt’s insistence on explicit continuous-time modeling rather than discrete-time curve fitting. 

---

### References (BibTeX)

```bibtex
@article{Shepherd1965,
  title        = {Design of Primary and Secondary Cells: Part 2. An Equation Describing Battery Discharge},
  author       = {Shepherd, C. M.},
  journal      = {Journal of the Electrochemical Society},
  volume       = {112},
  number       = {7},
  pages        = {657--664},
  year         = {1965},
  doi          = {10.1149/1.2423659}
}

@article{TremblayDessaint2009,
  title        = {Experimental Validation of a Battery Dynamic Model for EV Applications},
  author       = {Tremblay, Olivier and Dessaint, Louis-A.},
  journal      = {World Electric Vehicle Journal},
  volume       = {3},
  number       = {2},
  pages        = {289--298},
  year         = {2009},
  doi          = {10.3390/wevj3020289}
}

@article{Plett2004,
  title        = {Extended Kalman Filtering for Battery Management Systems of LiPB-Based HEV Battery Packs: Part 1. Background},
  author       = {Plett, Gregory L.},
  journal      = {Journal of Power Sources},
  volume       = {134},
  number       = {2},
  pages        = {252--261},
  year         = {2004},
  doi          = {10.1016/j.jpowsour.2004.02.031}
}

@article{DoyleFullerNewman1993,
  title        = {Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell},
  author       = {Doyle, Marc and Fuller, Thomas F. and Newman, John},
  journal      = {Journal of the Electrochemical Society},
  volume       = {140},
  number       = {6},
  pages        = {1526--1533},
  year         = {1993},
  doi          = {10.1149/1.2221597}
}
```
