## Model Formulation and Solution

### 1. Mechanistic Narrative for “Unpredictable” Battery Life

Battery-life “unpredictability” is not treated as randomness by fiat; it emerges from a **closed-loop nonlinear dynamical system** driven by time-varying user behavior. Three mechanisms dominate:

1. **Uncertain, time-varying inputs**: screen brightness (L(t)), processor load (C(t)), network activity (N(t)), signal quality (\Psi(t)), and ambient temperature (T_a(t)) fluctuate continuously, inducing a fluctuating power request (P_{\mathrm{tot}}(t)).

2. **Constant-power-load (CPL) nonlinearity**: smartphones behave approximately as CPLs at short time scales; thus the discharge current (I(t)) is not prescribed but must satisfy (P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)I(t)). As the terminal voltage declines (low SOC, cold temperature, polarization), the required current increases disproportionately, accelerating depletion.

3. **State memory**: polarization (v_p(t)) and temperature (T_b(t)) store information about the recent past; therefore, identical “current usage” can drain differently depending on what happened minutes earlier (gaming burst, radio tail, or cold exposure).

This narrative is included explicitly so that every equation below has a clear physical role in the causal chain
[
(L,C,N,\Psi,T_a)\ \Rightarrow\ P_{\mathrm{tot}}\ \Rightarrow\ I\ \Rightarrow\ (z,v_p,T_b,S)\ \Rightarrow\ V_{\mathrm{term}},\ \mathrm{TTE}.
]

---

### 2. State Variables, Inputs, and Outputs

#### 2.1 State vector

We model the battery–phone system as a continuous-time state-space system with
[
\mathbf{x}(t)=\big[z(t),,v_p(t),,T_b(t),,S(t),,w(t)\big]^\top,
]
where

* (z(t)\in[0,1]): state of charge (SOC).
* (v_p(t)) (V): polarization voltage (electrochemical transient “memory”).
* (T_b(t)) (K): battery temperature.
* (S(t)\in(0,1]): state of health (SOH), interpreted as retained capacity fraction.
* (w(t)\in[0,1]): radio “tail” activation level (continuous surrogate of network high-power persistence).

#### 2.2 Inputs (usage profile)

[
\mathbf{u}(t)=\big[L(t),,C(t),,N(t),,\Psi(t),,T_a(t)\big]^\top,
]
where (L,C,N\in[0,1]), signal quality (\Psi(t)\in(0,1]) (larger means better), and (T_a(t)) is ambient temperature.

#### 2.3 Outputs

* Terminal voltage (V_{\mathrm{term}}(t))
* SOC (z(t))
* Time-to-empty (\mathrm{TTE}) defined via a voltage cutoff and feasibility conditions (Section 6)

---

### 3. Equivalent Circuit and Core Electro–Thermal–Aging Dynamics

#### 3.1 Terminal voltage: 1st-order Thevenin ECM

We use a first-order Thevenin equivalent circuit with one polarization branch:
[
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}\big(z(t)\big)-v_p(t)-I(t),R_0\big(T_b(t),S(t)\big).
]
This model is a practical compromise: it captures nonlinear voltage behavior and transient polarization while remaining identifiable and computationally efficient.

#### 3.2 SOC dynamics (charge conservation)

Let (Q_{\mathrm{eff}}(T_b,S)) be the effective deliverable capacity (Ah). Then
[
\boxed{
\frac{dz}{dt}=-\frac{I(t)}{3600,Q_{\mathrm{eff}}\big(T_b(t),S(t)\big)}.
}
]
The factor (3600) converts Ah to Coulombs.

#### 3.3 Polarization dynamics (RC memory)

[
\boxed{
\frac{dv_p}{dt}=\frac{I(t)}{C_1}-\frac{v_p(t)}{R_1C_1}.
}
]
The time constant (\tau_p=R_1C_1) governs relaxation after workload changes.

#### 3.4 Thermal dynamics (lumped energy balance)

[
\boxed{
\frac{dT_b}{dt}=\frac{1}{C_{\mathrm{th}}}\Big(I(t)^2R_0(T_b,S)+I(t),v_p(t)-hA\big(T_b(t)-T_a(t)\big)\Big).
}
]

* (I^2R_0): ohmic heating
* (Iv_p): polarization heat
* (hA(T_b-T_a)): convective cooling
* (C_{\mathrm{th}}): effective thermal capacitance

#### 3.5 SOH dynamics: explicit long-horizon mechanism (SEI-inspired)

Even though (\Delta S) is small during a single discharge, writing a dynamical SOH equation signals mechanistic completeness and enables multi-cycle forecasting.

**Option A (compact throughput + Arrhenius):**
[
\boxed{
\frac{dS}{dt}=-\lambda_{\mathrm{sei}},|I(t)|^{m}\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b(t)}\right),
\qquad 0\le m\le 1.
}
]

**Option B (explicit SEI thickness state, diffusion-limited growth):**
Introduce SEI thickness (\delta(t)) and define
[
\frac{d\delta}{dt}
==================

k_{\delta},|I(t)|^{m}\exp!\left(-\frac{E_{\delta}}{R_gT_b}\right)\frac{1}{\delta+\delta_0},
\qquad
\frac{dS}{dt}=-\eta_{\delta},\frac{d\delta}{dt}.
]
For Question 1 (single discharge), Option A is typically sufficient and numerically lighter; Option B is presented as an upgrade path for multi-cycle study.

---

### 4. Multiphysics Power Mapping: (L,C,N,\Psi\rightarrow P_{\mathrm{tot}}(t))

Smartphones can be modeled as a sum of component power demands. We define
[
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}\big(L(t)\big)+P_{\mathrm{cpu}}\big(C(t)\big)+P_{\mathrm{net}}\big(N(t),\Psi(t),w(t)\big).
]

#### 4.1 Screen power

A smooth brightness response is captured by
[
\boxed{
P_{\mathrm{scr}}(L)=P_{\mathrm{scr},0}+k_L,L^{\gamma},\qquad \gamma>1.
}
]
This form conveniently supports OLED/LCD scenario analysis: OLED-like behavior tends to have stronger convexity (larger effective (\gamma)).

#### 4.2 CPU power (DVFS-consistent convexity)

A minimal DVFS-consistent convex map is
[
\boxed{
P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu},0}+k_C,C^{\eta},\qquad \eta>1,
}
]
reflecting that CPU power often grows faster than linearly with load due to frequency/voltage scaling.

#### 4.3 Network power with signal-quality penalty and radio tail

We encode weak-signal amplification via a power law and include a continuous tail state:
[
\boxed{
P_{\mathrm{net}}(N,\Psi,w)=P_{\mathrm{net},0}+k_N,\frac{N}{(\Psi+\varepsilon)^{\kappa}}+k_{\mathrm{tail}},w,
\qquad \kappa>0.
}
]

**Tail-state dynamics (continuous surrogate of radio persistence):**
[
\boxed{
\frac{dw}{dt}=\frac{\sigma(N(t))-w(t)}{\tau(N(t))},
\qquad
\tau(N)=
\begin{cases}
\tau_{\uparrow}, & \sigma(N)\ge w,\
\tau_{\downarrow}, & \sigma(N)< w,
\end{cases}
}
]
with (\tau_{\uparrow}\ll\tau_{\downarrow}) capturing fast activation and slow decay; (\sigma(\cdot)) may be (\sigma(N)=\min{1,N}). This introduces memory without discrete state machines, keeping the overall model continuous-time.

---

### 5. Current Closure Under Constant-Power Load (CPL)

#### 5.1 Algebraic closure

We impose the CPL constraint
[
\boxed{
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t),I(t).
}
]
Substituting (V_{\mathrm{term}}=V_{\mathrm{oc}}-v_p-I R_0) yields
[
R_0 I^2-\big(V_{\mathrm{oc}}(z)-v_p\big)I+P_{\mathrm{tot}}=0.
]

#### 5.2 Physically admissible current (quadratic root)

[
\boxed{
I(t)=\frac{V_{\mathrm{oc}}(z)-v_p-\sqrt{\Delta(t)}}{2R_0(T_b,S)},
\quad
\Delta(t)=\big(V_{\mathrm{oc}}(z)-v_p\big)^2-4R_0(T_b,S),P_{\mathrm{tot}}(t).
}
]
We take the smaller root to maintain (V_{\mathrm{term}}\ge 0) and avoid unphysical large currents.

#### 5.3 Feasibility / collapse condition

[
\Delta(t)\ge 0
]
is required for real (I(t)). If (\Delta(t)\le 0), the requested power exceeds deliverable power at that state; the phone effectively shuts down (voltage collapse), which provides a mechanistic explanation for “sudden drops” under cold/low SOC/weak signal.

---

### 6. Constitutive Relations: (V_{\mathrm{oc}}(z)), (R_0(T_b,S)), (Q_{\mathrm{eff}}(T_b,S))

#### 6.1 Open-circuit voltage: modified Shepherd form

[
\boxed{
V_{\mathrm{oc}}(z)=E_0-K\left(\frac{1}{z}-1\right)+A,e^{-B(1-z)}.
}
]
This captures the plateau and the end-of-discharge knee smoothly.

#### 6.2 Internal resistance: Arrhenius temperature dependence + SOH correction

[
\boxed{
R_0(T_b,S)=R_{\mathrm{ref}}
\exp!\left[\frac{E_a}{R_g}\left(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\right)\right]\Big(1+\eta_R(1-S)\Big).
}
]
Cold increases (R_0); aging (lower (S)) increases resistance.

#### 6.3 Effective capacity: temperature + aging

[
\boxed{
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}},S\Big[1-\alpha_Q,(T_{\mathrm{ref}}-T_b)\Big]*+,
}
]
where ([\cdot]*+=\max(\cdot,\kappa_{\min})) prevents nonphysical negative capacity.

---

### 7. Final Closed System (ODE + algebraic current)

Collecting Sections 3–6, the model is a nonlinear ODE system driven by (\mathbf{u}(t)), with a nested algebraic solver for (I(t)):
[
\dot{\mathbf{x}}(t)=\mathbf{f}\big(t,\mathbf{x}(t),\mathbf{u}(t)\big),
\quad
I(t)=\mathcal{I}\big(\mathbf{x}(t),\mathbf{u}(t)\big)
]
where (\mathcal{I}) is the quadratic-root mapping.

**Initial conditions (must be stated explicitly):**
[
z(0)=z_0,\quad v_p(0)=0,\quad T_b(0)=T_a(0),\quad S(0)=S_0,\quad w(0)=0.
]

---

### 8. Parameter Estimation (Hybrid: literature + identifiable fits)

A fully free fit is ill-posed; we use a **hybrid identification** strategy:

#### 8.1 Literature / specification parameters

* (Q_{\mathrm{nom}}), nominal voltage class, plausible cutoff (V_{\mathrm{cut}})
* thermal scales (C_{\mathrm{th}},hA) in reasonable ranges for compact devices
* activation energies (E_a,E_{\mathrm{sei}}) as literature-consistent order-of-magnitude

#### 8.2 OCV curve fit: ((E_0,K,A,B))

From quasi-equilibrium OCV–SOC samples ({(z_i,V_i)}):
[
\min_{E_0,K,A,B}\sum_i\left[V_i - V_{\mathrm{oc}}(z_i)\right]^2,
\quad E_0,K,A,B>0.
]

#### 8.3 Pulse identification: (R_0,R_1,C_1)

Apply a current pulse (\Delta I). The instantaneous voltage drop estimates
[
R_0\approx \frac{\Delta V(0^+)}{\Delta I}.
]
The relaxation yields (\tau_p=R_1C_1) from exponential decay; (R_1) from amplitude and (C_1=\tau_p/R_1).

#### 8.4 Signal exponent (\kappa) (or exponential alternative)

From controlled network tests at fixed throughput (N) with varying (\Psi), fit:
[
\ln\big(P_{\mathrm{net}}-P_{\mathrm{net},0}-k_{\mathrm{tail}}w\big)
===================================================================

\ln(k_NN)-\kappa \ln(\Psi+\varepsilon).
]

---

### 9. Scenario Simulation (Synthetic yet physics-plausible)

We choose a representative smartphone battery:

* (Q_{\mathrm{nom}}=4000,\mathrm{mAh}=4,\mathrm{Ah})
* nominal voltage (\approx 3.7,\mathrm{V})

#### 9.1 A realistic alternating-load usage profile

Define a 6-hour profile with alternating low/high intensity segments. A smooth transition operator avoids discontinuities:
[
\mathrm{win}(t;a,b,\delta)=\frac{1}{1+e^{-(t-a)/\delta}}-\frac{1}{1+e^{-(t-b)/\delta}}.
]
Then
[
L(t)=\sum_j L_j,\mathrm{win}(t;a_j,b_j,\delta),\quad
C(t)=\sum_j C_j,\mathrm{win}(t;a_j,b_j,\delta),\quad
N(t)=\sum_j N_j,\mathrm{win}(t;a_j,b_j,\delta),
]
with (\delta\approx 20) s.

Example segment levels (normalized):

* standby/messaging: (L=0.10, C=0.10, N=0.20)
* streaming: (L=0.70, C=0.40, N=0.60)
* gaming: (L=0.90, C=0.90, N=0.50)
* navigation: (L=0.80, C=0.60, N=0.80)
  Signal quality (\Psi(t)) can be set to “good” for most intervals, with one “poor-signal” hour to test the (\Psi^{-\kappa}) mechanism.

---

### 10. Numerical Solution

#### 10.1 RK4 with nested algebraic current solve

We integrate the ODEs using classical RK4. At each substage, we recompute:
[
P_{\mathrm{tot}}\rightarrow V_{\mathrm{oc}}\rightarrow R_0,Q_{\mathrm{eff}}\rightarrow \Delta \rightarrow I
]
and then evaluate (\dot{\mathbf{x}}).

**Algorithm 1 (RK4 + CPL closure)**

1. Given (\mathbf{x}_n) at time (t_n), compute inputs (\mathbf{u}(t_n)).
2. Compute (P_{\mathrm{tot}}(t_n)) and solve (I(t_n)) from the quadratic root.
3. Evaluate RK4 stages (\mathbf{k}_1,\dots,\mathbf{k}_4), solving (I) inside each stage.
4. Update (\mathbf{x}_{n+1}).
5. Stop if (V_{\mathrm{term}}\le V_{\mathrm{cut}}) or (z\le 0) or (\Delta\le 0).

#### 10.2 Step size, stability, and convergence criterion

Let (\tau_p=R_1C_1). Choose
[
\Delta t \le 0.05,\tau_p
]
to resolve polarization. Perform step-halving verification:
[
|z_{\Delta t}-z_{\Delta t/2}|_\infty < \varepsilon_z,\quad \varepsilon_z=10^{-4}.
]
Report that predicted TTE changes by less than a chosen tolerance (e.g., 1%) when halving (\Delta t).

---

### 11. Result Presentation (what to report in the paper)

#### 11.1 Primary plots

* (z(t)) (SOC curve), with shaded regions indicating usage segments
* (I(t)) and (P_{\mathrm{tot}}(t)) (secondary axis)
* (T_b(t)) to show thermal feedback
* Optional: (\Delta(t)) to visualize proximity to voltage collapse under weak signal/cold

#### 11.2 Key scalar outputs

* (\mathrm{TTE}) under baseline (T_a=25^\circ\mathrm{C})
* (\mathrm{TTE}) under cold (T_a=0^\circ\mathrm{C}) and hot (T_a=40^\circ\mathrm{C})
* Sensitivity of TTE to (\Psi) (good vs poor signal), holding (N) fixed

---

### 12. Discussion: sanity checks tied to physics

* **Energy check**: a (4,\mathrm{Ah}), (3.7,\mathrm{V}) battery stores (\approx 14.8,\mathrm{Wh}); if average (P_{\mathrm{tot}}) is (2.5,\mathrm{W}), a (5\text{–}7) hour TTE is plausible.
* **Cold penalty**: (R_0\uparrow) and (Q_{\mathrm{eff}}\downarrow) shorten TTE.
* **Weak signal penalty**: when (N) is significant, (\Psi^{-\kappa}) materially increases (P_{\mathrm{tot}}), pushing (\Delta) toward zero and shortening TTE.
* **Memory effects**: bursts elevate (v_p) and (w), causing post-burst drain that would not appear in static models.

---

## 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},
  year    = {1965},
  volume  = {112},
  number  = {7},
  pages   = {657--664}
}

@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},
  year    = {2009},
  volume  = {3},
  number  = {2},
  pages   = {289--298}
}

@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},
  year    = {2004},
  volume  = {134},
  number  = {2},
  pages   = {252--261}
}
```