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# Front Matter封面与前置页
## Title题目
**A Mechanism-Driven Continuous-Time Model for Smartphone Battery Drain Under Constant-Power Loads: Component Power Mapping, Electro-Thermal-Aging Coupling, and Feasibility-Based Shutdown Prediction**
(若你们中文论文:
**基于恒功率负载闭环的智能手机电池连续时间机理模型:功耗分解、热-电-老化耦合与可行性掉电判据**
---
## Team Information队伍信息按比赛模板填写
* **Team Control Number:** [填写]
* **School/Institution:** [填写]
* **Team Members:** [填写]
* **Date:** [填写]
> 注:这一块通常由比赛提交模板决定,你只要把占位符替换成官方要求格式即可。
---
## Abstract摘要
Smartphone runtime is governed by multi-source, time-varying power demands from the screen, CPU, and wireless communication, and it often exhibits nonlinear behaviors such as abrupt shutdown at low state-of-charge (SOC), low temperature, or advanced aging. To capture these mechanisms, we develop a continuous-time, physics-informed model featuring a state vector (\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top), where (z) is SOC, (v_p) is polarization voltage (memory), (T_b) is battery temperature, (S) is state-of-health (SOH), and (w) represents a continuous network “tail” state. Exogenous inputs (\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top) describe screen brightness, CPU load, network activity, signal quality, and ambient temperature, respectively. Total power demand is decomposed explicitly into screen/CPU/network components, with the network term incorporating a signal-quality penalty and tail dynamics. On the battery side, a first-order equivalent circuit model (ECM) is coupled to the load through a constant power load (CPL) closure, yielding a nonlinear currentvoltage feedback and a feasibility discriminant (\Delta(t)\ge 0) that explains voltage collapse and sudden shutdown. Temperature- and SOH-dependent internal resistance and effective capacity are included via Arrhenius and capacity-scaling relations, while a compact SEI-inspired degradation law governs SOH evolution. For robustness and device realism, we add three lightweight refinements: (i) a low-SOC regularization in the OCV model, (ii) a nonnegative polarization heat formulation, and (iii) a temperature-dependent current cap representing OS/PMIC throttling. The resulting framework supports numerical simulation, time-to-empty (TTE) prediction, uncertainty quantification, and actionable power-management recommendations.
---
## Keywords关键词
Smartphone battery drain; constant power load (CPL); equivalent circuit model (ECM); electro-thermal coupling; battery aging (SOH); network tail energy; feasibility discriminant; time-to-empty (TTE)
---
# Summary SheetMCM 一页摘要页 / Executive Summary
> **说明**:这一页要“像海报一样快读”。下面版本是可直接交稿的结构;你们跑完仿真后把括号内结果补上即可。
## Problem
We are asked to model smartphone battery drain in continuous time under realistic, time-varying usage. The model must predict battery terminal voltage and SOC evolution and determine the time-to-empty (TTE), while explaining nonlinear shutdown behaviors (e.g., abrupt power-off before SOC reaches zero) under adverse conditions such as poor signal quality, low temperature, and aging.
## Model Overview
**States and inputs.** We define the state vector
[
\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top,
]
where (z) is SOC, (v_p) is polarization voltage, (T_b) is battery temperature, (S) is SOH, and (w) is the continuous network tail state. Inputs are
[
\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top,
]
describing brightness, CPU load, network activity, signal quality, and ambient temperature.
**Component-level power mapping.** Total demanded power is decomposed as
[
P_{\mathrm{tot}}=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L)+P_{\mathrm{cpu}}(C)+P_{\mathrm{net}}(N,\Psi,w),
]
with superlinear screen/CPU mappings and an explicit signal-quality penalty plus tail term in the network power.
**Battery dynamics and CPL closure.** A first-order ECM gives terminal voltage
[
V_{\mathrm{term}}=V_{\mathrm{oc}}(z)-v_p-I R_0(T_b,S).
]
The load is modeled as a constant power load (CPL),
[
P_{\mathrm{tot}}=V_{\mathrm{term}}I,
]
leading to a quadratic current solution and a feasibility discriminant
[
\Delta=(V_{\mathrm{oc}}-v_p)^2-4R_0P_{\mathrm{tot}}.
]
When (\Delta<0), maintaining the demanded power becomes infeasible, providing a mechanism for voltage collapse and abrupt shutdown.
**Electro-thermal-aging coupling.** SOC, polarization, temperature, and SOH evolve via coupled ODEs (including Arrhenius resistance, temperature/SOH-dependent effective capacity, and an SEI-inspired SOH decay law). Network tail energy is captured by a continuous-time tail state (w(t)).
**Robustness refinements (lightweight, non-invasive).**
1. Low-SOC regularization in OCV using (z_{\mathrm{eff}}=\max(z,z_{\min})) to avoid singularity.
2. Nonnegative polarization heat via (v_p^2/R_1) in the thermal source term.
3. A temperature-dependent current cap (I=\min(I_{\mathrm{CPL}},I_{\max}(T_b))) to represent OS/PMIC throttling.
## Numerical Method
We solve the coupled ODEs using RK4 (or an adaptive RungeKutta method) with a nested algebraic current evaluation at each substep. Step size is constrained by the polarization time constant (\tau_p=R_1C_1), and convergence is verified by step-halving until (|z_{\Delta t}-z_{\Delta t/2}|_\infty<10^{-4}), with TTE changes below 1%.
## Key Results (to be filled with your simulations)
* **Baseline runtime (TTE):** mean (\approx) [***] h, median (\approx) [***] h, 5th95th percentile ([***],[***]) h under the baseline usage scenario.
* **Sudden shutdown mechanism:** infeasibility events ((\Delta<0)) occur primarily when [high demand + elevated (R_0)] coincide (e.g., weak signal (\Psi\downarrow), low (T_b), low (S)), precipitating rapid voltage collapse.
* **Impact of throttling (current cap):** applying (I_{\max}(T_b)) increases the 5th-percentile TTE by approximately [***]%, and reduces infeasibility/shutdown-risk events by [***]%.
* **Sensitivity (Sobol):** the largest total-effect indices are associated with [(k_N,\kappa)] under weak-signal regimes and with [(k_L,\gamma)] under high-brightness usage; ambient temperature (T_a) shows strong interaction effects via (R_0(T_b,S)) and (Q_{\mathrm{eff}}(T_b,S)).
## Conclusions
We present a mechanism-driven continuous-time smartphone battery model that unifies (i) component-level power demand with explicit signal-quality effects and network tail energy, (ii) an ECM battery model coupled through a CPL closure, and (iii) electro-thermal-aging interactions. The feasibility discriminant (\Delta) provides an interpretable explanation for abrupt shutdown behaviors beyond simple SOC depletion.
## Recommendations
* **User-level:** reduce brightness (L) and avoid sustained high-throughput activity (N) in poor signal conditions ((\Psi) low) to mitigate network power amplification and tail energy.
* **System-level (OS/PMIC):** implement adaptive power caps or temperature-dependent current limits to prevent CPL-driven current escalation at low voltage/high resistance, thereby improving worst-case runtime and reducing collapse risk.
* **Network-level:** tail-state-aware scheduling (batching transmissions) can reduce (w(t)) and tail energy, improving TTE with minimal user impact.
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%========================================================
\section{Problem Restatement \& Objectives}
\label{sec:problem}
\subsection{Restatement of the Problem}
\label{subsec:restatement}
The 2026 MCM Problem A concerns continuous-time prediction of smartphone battery drain under time-varying usage.
A smartphone is subject to multiple power-consuming components---most prominently the display, CPU workload, and cellular/Wi-Fi communication---whose intensities evolve over time according to user behavior and network conditions.
Meanwhile, the battery exhibits coupled electro-thermal dynamics and gradual health degradation.
The task is to construct a mechanism-driven, continuous-time model that maps future usage profiles to battery states and terminal voltage, and then to estimate the remaining operating time before the device shuts down.
In particular, given (measured, prescribed, or scenario-generated) time series describing user/device usage and ambient conditions, we aim to:
(i) predict the trajectories of key battery states (e.g., state-of-charge and temperature) and the terminal voltage;
(ii) compute the time-to-empty (TTE) defined by physically meaningful shutdown criteria; and
(iii) interpret sudden shutdown phenomena through explicit feasibility mechanisms rather than black-box regression.
\subsection{Inputs, Outputs, and Prediction Tasks}
\label{subsec:io_tasks}
We represent the battery-in-phone system by a state vector
\[
\mathbf{x}(t)=[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)]^\top,
\]
where \(z\) is state-of-charge (SOC), \(v_p\) is polarization voltage (memory effect of the ECM),
\(T_b\) is battery temperature, \(S\) is state-of-health (SOH, capacity fraction), and \(w\) is the radio tail state.
The exogenous inputs are collected as
\[
\mathbf{u}(t)=[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)]^\top,
\]
where \(L\in[0,1]\) denotes normalized screen brightness,
\(C\in[0,1]\) the normalized CPU load,
\(N\in[0,1]\) the normalized network activity level (throughput/airtime proxy),
\(\Psi>0\) a signal-quality indicator (larger is better),
and \(T_a\) the ambient temperature.
\paragraph{Primary predicted outputs.}
The model produces the battery terminal voltage and SOC trajectories,
\[
V_{\mathrm{term}}(t), \qquad z(t),
\]
as well as the time-to-empty (TTE), defined as the first time the device becomes inoperable under the specified shutdown criteria:
\[
\mathrm{TTE}=\inf\Big\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\Big\}.
\]
Here \(V_{\mathrm{cut}}\) is the cutoff voltage dictated by system protection (BMS/PMIC).
\paragraph{Prediction tasks.}
Given \(\mathbf{x}(0)\) and a future input profile \(\mathbf{u}(t)\) on a horizon \([0,T]\), the prediction tasks are:
\begin{enumerate}
\item \textbf{State/voltage forecasting:} compute \(\mathbf{x}(t)\) and \(V_{\mathrm{term}}(t)\) for \(t\in[0,T]\);
\item \textbf{Runtime estimation:} compute \(\mathrm{TTE}\) from the stopping rule above;
\item \textbf{Mechanistic interpretation:} attribute shutdown to depletion (\(z\to 0\)) or voltage protection (\(V_{\mathrm{term}}\le V_{\mathrm{cut}}\)), and quantify risk of power infeasibility (Section~\ref{subsec:metrics_scenarios}).
\end{enumerate}
\subsection{Performance Metrics and Usage-Scenario Description}
\label{subsec:metrics_scenarios}
\paragraph{Operational termination and reliability-oriented metrics.}
The principal performance metric is the operating time before shutdown, \(\mathrm{TTE}\).
For evaluation and comparison across scenarios, we also report:
\begin{itemize}
\item \textbf{Terminal-voltage margin:} \(\min_{t\in[0,\mathrm{TTE}]}(V_{\mathrm{term}}(t)-V_{\mathrm{cut}})\), which indicates how close the device operates to the cutoff boundary;
\item \textbf{Delivered-energy proxy:} \(E_{\mathrm{del}}=\int_{0}^{\mathrm{TTE}} V_{\mathrm{term}}(t)I(t)\,dt\) (when current \(I(t)\) is available from the closure), which supports sanity checks against SOC depletion;
\item \textbf{Thermal exposure:} \(\max_{t\in[0,\mathrm{TTE}]} T_b(t)\), reflecting potential thermal throttling or safety constraints.
\end{itemize}
\paragraph{Risk event: CPL feasibility (voltage-collapse risk).}
Because the load is modeled as a constant-power demand (CPL) coupled to the electrochemical model, a feasibility condition naturally arises.
Let
\[
\Delta(t)=\big(V_{\mathrm{oc}}(z(t)) - v_p(t)\big)^2 - 4R_0(T_b(t),S(t))\,P_{\mathrm{tot}}(t),
\]
where \(P_{\mathrm{tot}}(t)\) is the demanded total power and \(R_0\) the ohmic resistance.
When \(\Delta(t)<0\), the CPL algebraic closure admits no real current solution, indicating that the demanded power is infeasible given the instantaneous battery capability and may lead to abrupt voltage collapse.
We therefore introduce the \emph{first risk time}
\[
t_{\Delta}=\inf\{t>0:\ \Delta(t)\le 0\},
\]
as an auxiliary diagnostic.
In later sections, we use \(t_\Delta\) to distinguish \emph{infeasibility-driven} shutdown risk from ordinary energy depletion.
\paragraph{Representative usage scenarios.}
To ensure that conclusions are interpretable and reproducible, we evaluate the model under a small set of canonical usage scenarios, each defined by a characteristic input pattern \(\mathbf{u}(t)\).
Table~\ref{tab:scenarios} summarizes the scenarios used throughout the paper.
\begin{table}[t]
\centering
\caption{Representative usage scenarios and their qualitative input characteristics.}
\label{tab:scenarios}
\begin{tabular}{p{2.4cm}p{10.6cm}}
\hline
Scenario & Input characteristics \(\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top\) \\
\hline
Standby/Idle &
Low brightness \(L\approx 0\) (screen off), low CPU \(C\ll 1\), sporadic network \(N\approx 0\) with residual tail \(w\), typical \(\Psi\), moderate \(T_a\). \\
Browsing/Social &
Moderate \(L\), moderate CPU \(C\), intermittent network bursts \(N(t)\) with tail effects, typical-to-good \(\Psi\), moderate \(T_a\). \\
Video Streaming &
High \(L\), sustained moderate CPU \(C\), sustained network activity \(N\) (downlink), sensitivity to \(\Psi\); moderate \(T_a\). \\
Gaming/High Compute &
High \(L\), high CPU \(C\) (near saturation), moderate network \(N\), typical \(\Psi\); emphasizes thermal rise and possible throttling. \\
Weak Signal (Stress) &
Moderate-to-high \(L\), moderate CPU \(C\), nontrivial \(N\) under poor signal \(\Psi\downarrow\); stresses the signal-quality penalty in \(P_{\mathrm{net}}(N,\Psi,w)\) and increases collapse risk. \\
Cold Ambient (Stress) &
Any of the above with low \(T_a\); highlights increased \(R_0\) and reduced \(Q_{\mathrm{eff}}\), potentially shortening TTE and increasing \(t_\Delta\) likelihood. \\
\hline
\end{tabular}
\end{table}
The above scenarios are not tied to a specific dataset; they can be instantiated using recorded traces or generated synthetically (e.g., piecewise-smooth profiles or stochastic processes) while keeping the same physical meaning of each input channel.
This design supports both deterministic simulations and uncertainty quantification (Monte Carlo) in later sections.
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% =========================================================
% Section 3: Nomenclature (Symbols and Variables)
% This block is self-contained and can be pasted into the paper.
% =========================================================
\section{Nomenclature: Symbols and Variables}\label{sec:nomenclature}
To ensure clarity and reproducibility, this section summarizes the state variables, exogenous inputs, model outputs, derived quantities, and the parameter set used throughout the paper. All symbols are consistent with the mechanistic continuous-time framework defined in Sections~\ref{sec:model}--\ref{sec:numerics}.
\subsection{State Vector $\mathbf{x}(t)$}\label{subsec:state}
We define the state vector
\begin{equation}
\mathbf{x}(t)=\big[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)\big]^\top.
\end{equation}
\begin{table}[h!]
\centering
\caption{State variables in $\mathbf{x}(t)$.}\label{tab:states}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Typical range / unit \\
\hline
$z(t)$ & State of charge (SOC) & $[0,1]$ \\
$v_p(t)$ & Polarization voltage (RC memory state) & V \\
$T_b(t)$ & Battery (cell) temperature & K \\
$S(t)$ & State of health (SOH), capacity fraction & $[0,1]$ \\
$w(t)$ & Radio tail state (continuous tail activity) & $[0,1]$ \\
\hline
\end{tabular}
\end{table}
\subsection{Input Vector $\mathbf{u}(t)$}\label{subsec:input}
The exogenous usage/environment inputs are
\begin{equation}
\mathbf{u}(t)=\big[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)\big]^\top.
\end{equation}
\begin{table}[h!]
\centering
\caption{Inputs in $\mathbf{u}(t)$.}\label{tab:inputs}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Typical range / unit \\
\hline
$L(t)$ & Screen brightness (normalized) & $[0,1]$ \\
$C(t)$ & CPU load (normalized) & $[0,1]$ \\
$N(t)$ & Network activity / throughput proxy (normalized) & $[0,1]$ \\
$\Psi(t)$ & Signal quality (larger is better) & dimensionless or normalized \\
$T_a(t)$ & Ambient temperature & K \\
\hline
\end{tabular}
\end{table}
\subsection{Outputs and Derived Quantities}\label{subsec:outputs-derived}
The primary outputs are the terminal voltage $V_{\mathrm{term}}(t)$, the SOC $z(t)$, and the time-to-empty (TTE). In addition, several derived quantities are used to couple the load-side power demand to the battery-side electro-thermal dynamics.
\paragraph{(i) Total power demand.}
The total power consumption is decomposed into background, screen, CPU, and networking components:
\begin{equation}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)),
\label{eq:Ptot_def}
\end{equation}
where
\begin{align}
P_{\mathrm{scr}}(L)&=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1, \label{eq:Pscr_def}\\
P_{\mathrm{cpu}}(C)&=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1, \label{eq:Pcpu_def}\\
P_{\mathrm{net}}(N,\Psi,w)&=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,\qquad \kappa>0. \label{eq:Pnet_def}
\end{align}
\paragraph{(ii) Terminal voltage.}
The battery terminal voltage is given by the first-order equivalent circuit model (ECM):
\begin{equation}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z(t)) - v_p(t) - I(t)\,R_0(T_b(t),S(t)).
\label{eq:Vterm_def}
\end{equation}
\paragraph{(iii) CPL feasibility discriminant.}
Under the constant-power-load (CPL) closure $P_{\mathrm{tot}}=V_{\mathrm{term}}I$, the algebraic current solve yields the discriminant
\begin{equation}
\Delta(t)=\big(V_{\mathrm{oc}}(z(t))-v_p(t)\big)^2-4\,R_0(T_b(t),S(t))\,P_{\mathrm{tot}}(t).
\label{eq:Delta_def}
\end{equation}
The CPL current is real-valued only if $\Delta(t)\ge 0$. When $\Delta(t)<0$, the requested power is infeasible given the instantaneous electrochemical state, which corresponds to a voltage-collapse risk event in the model.
\paragraph{(iv) Time-to-empty (TTE).}
The operational end time is defined by the earliest occurrence among voltage cutoff and SOC depletion:
\begin{equation}
\mathrm{TTE}=\inf\left\{t>0:\;V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\right\}.
\label{eq:TTE_def}
\end{equation}
\begin{table}[h!]
\centering
\caption{Key outputs and derived quantities.}\label{tab:derived}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Unit \\
\hline
$V_{\mathrm{term}}(t)$ & Terminal voltage & V \\
$I(t)$ & Battery current (discharge positive) & A \\
$P_{\mathrm{tot}}(t)$ & Total power demand & W \\
$V_{\mathrm{oc}}(z)$ & Open-circuit voltage (OCV) & V \\
$\Delta(t)$ & CPL feasibility discriminant & V$^2$ \\
$\mathrm{TTE}$ & Time-to-empty (operation end time) & s (or min, h) \\
$V_{\mathrm{cut}}$ & Voltage cutoff threshold & V \\
\hline
\end{tabular}
\end{table}
\subsection{Parameter Set and Units}\label{subsec:params}
Let $\Theta$ denote the full parameter set. For transparency, we group parameters by subsystem: load-side power mapping, ECM, thermal, and aging. Parameters may be identified from pulse tests, OCV--SOC curves, and device-level power measurements as described in Section~\ref{sec:numerics}.
\paragraph{(a) Power mapping parameters.}
\begin{table}[h!]
\centering
\caption{Power mapping parameters (load-side).}\label{tab:params_power}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$P_{\mathrm{bg}}$ & Background power & W & idle measurement \\
$P_{\mathrm{scr},0}$ & Screen baseline power & W & brightness sweep \\
$k_L$ & Screen power coefficient & W & brightness sweep \\
$\gamma$ & Screen superlinearity exponent & -- & brightness sweep \\
$P_{\mathrm{cpu},0}$ & CPU baseline power & W & CPU micro-benchmark \\
$k_C$ & CPU power coefficient & W & CPU micro-benchmark \\
$\eta$ & CPU superlinearity exponent & -- & CPU micro-benchmark \\
$P_{\mathrm{net},0}$ & Network baseline power & W & network idle \\
$k_N$ & Network activity coefficient & W & fixed-throughput tests \\
$\kappa$ & Signal-quality penalty exponent & -- & $\log$--$\log$ fit vs $\Psi$ \\
$\varepsilon$ & Signal-quality regularizer & same as $\Psi$ & chosen small, prevents singularity \\
$k_{\mathrm{tail}}$ & Tail power coefficient & W & tail decay fit \\
$\tau_\uparrow,\tau_\downarrow$ & Tail rise/decay time constants & s & tail transient fit \\
\hline
\end{tabular}
\end{table}
\paragraph{(b) ECM and electrochemical parameters.}
\begin{table}[h!]
\centering
\caption{ECM/electrochemical parameters.}\label{tab:params_ecm}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$E_0,K,A,B$ & Modified Shepherd OCV parameters & (V, V, V, --) & OCV--SOC curve fit \\
$R_{\mathrm{ref}}$ & Reference ohmic resistance & $\Omega$ & pulse $\Delta V(0^+)/\Delta I$ \\
$E_a$ & Activation energy for $R_0(T)$ & J/mol & multi-$T$ resistance fit \\
$T_{\mathrm{ref}}$ & Reference temperature & K & fixed (e.g., 298 K) \\
$\eta_R$ & SOH-to-resistance coefficient & -- & multi-SOH resistance fit \\
$R_1$ & Polarization resistance & $\Omega$ & pulse relaxation \\
$C_1$ & Polarization capacitance & F & pulse relaxation \\
\hline
\end{tabular}
\end{table}
\paragraph{(c) Capacity and thermal parameters.}
\begin{table}[h!]
\centering
\caption{Capacity and thermal parameters.}\label{tab:params_thermal}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$Q_{\mathrm{nom}}$ & Nominal capacity & Ah & datasheet / capacity test \\
$\alpha_Q$ & Temperature-capacity coefficient & 1/K & multi-$T$ capacity test \\
$C_{\mathrm{th}}$ & Lumped thermal capacitance & J/K & heating transient fit \\
$hA$ & Effective heat transfer coefficient & W/K & cooling transient fit \\
\hline
\end{tabular}
\end{table}
\paragraph{(d) Aging (SOH) parameters.}
\begin{table}[h!]
\centering
\caption{SOH degradation parameters (SEI-driven compact model).}\label{tab:params_aging}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$\lambda_{\mathrm{sei}}$ & SEI degradation rate prefactor & s$^{-1}$A$^{-m}$ & aging dataset fit \\
$m$ & Current-stress exponent & -- & aging dataset fit \\
$E_{\mathrm{sei}}$ & SEI activation energy & J/mol & aging dataset fit \\
$R_g$ & Universal gas constant & J/(mol$\cdot$K) & constant \\
\hline
\end{tabular}
\end{table}
\paragraph{(e) Robustness/control micro-adjustments.}
The following quantities support numerical robustness and device-level throttling without altering the core mechanism:
\begin{equation}
z_{\min}\in(0,1)\ \text{(low-SOC guard for OCV evaluation)},\qquad
V_{\mathrm{cut}}\ \text{(shutdown voltage)},\qquad
I_{\max,0},\rho_T\ \text{(current limit parameters)}.
\end{equation}
Their calibration and usage are detailed in Section~\ref{sec:numerics}.
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\section{Assumptions}\label{sec:assumptions}
To balance physical fidelity, interpretability, and computational tractability, we adopt the following modeling assumptions. These assumptions are consistent with the continuous-time, mechanism-driven framework developed in Sections~\ref{sec:model_formulation}--\ref{sec:numerics} and are intended to match typical smartphone operating conditions.
\subsection{Structural Assumptions}\label{sec:assumptions_structural}
\begin{enumerate}
\item \textbf{Single-cell lumped equivalent.}
The battery pack is represented by an equivalent single cell with lumped electrical and thermal states. The terminal behavior is captured by a first-order equivalent circuit model (ECM) comprising open-circuit voltage (OCV), an ohmic resistance, and one polarization (RC) branch.
\item \textbf{Additive component power mapping.}
The device power demand is decomposed into additive contributions from background processes, display, CPU, and network subsystems:
\(
P_{\mathrm{tot}} = P_{\mathrm{bg}} + P_{\mathrm{scr}}(L) + P_{\mathrm{cpu}}(C) + P_{\mathrm{net}}(N,\Psi,w).
\)
Cross-couplings among subsystems (e.g., CPU--network interactions) are treated as second-order effects and are absorbed into the calibrated parameters of the component maps.
\item \textbf{Normalized inputs and bounded states.}
Usage inputs are normalized to dimensionless intensities \(L,C,N\in[0,1]\), and the radio-tail state satisfies \(w\in[0,1]\). State variables are interpreted physically and are constrained to admissible ranges (e.g., \(z\in[0,1]\), \(S\in[0,1]\)) up to numerical tolerances.
\end{enumerate}
\subsection{Load-Side Assumptions}\label{sec:assumptions_load}
\begin{enumerate}
\item \textbf{Constant-power load (CPL) closure.}
Over the modeling time scale, the smartphone power management system is approximated as imposing an instantaneous power demand \(P_{\mathrm{tot}}(t)\) at the battery terminals, i.e.,
\(
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t).
\)
This CPL closure is used to capture the key nonlinear feedback whereby decreasing terminal voltage can induce increasing current draw under fixed power demand.
\item \textbf{Feasibility interpretation via discriminant.}
The quadratic CPL relation yields a discriminant \(\Delta(t)\). When \(\Delta(t)<0\), sustaining the requested power with the current electrical state is infeasible, indicating a voltage-collapse risk. This provides a mechanistic explanation for ``sudden shutdown'' events observed in practice.
\item \textbf{Optional derating (current/power limiting).}
Smartphones typically derate performance (e.g., frequency throttling or PMIC current limiting) under low-voltage or high-temperature conditions. We therefore allow an optional saturation policy, e.g.,
\(
I(t)=\min\{I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\},
\)
which preserves the original CPL behavior when \(I_{\mathrm{CPL}}\le I_{\max}\) while enabling safe operation (at reduced delivered power) when the requested power would otherwise drive excessive current.
\end{enumerate}
\subsection{Thermal Assumptions}\label{sec:assumptions_thermal}
\begin{enumerate}
\item \textbf{Lumped thermal capacitance.}
The battery temperature is modeled by a single lumped node \(T_b(t)\) with effective thermal capacitance \(C_{\mathrm{th}}\). Spatial gradients within the cell or across the device chassis are neglected.
\item \textbf{Dominant heat sources and linear heat rejection.}
Heat generation is attributed to ohmic loss and polarization-branch dissipation, while heat rejection to the environment is modeled by linear convection/conduction:
\[
\dot T_b=\frac{1}{C_{\mathrm{th}}}\Big(I^2R_0+\frac{v_p^2}{R_1}-hA\,(T_b-T_a)\Big).
\]
Radiative effects and temperature dependence of \(hA\) are neglected over normal operating ranges.
\item \textbf{Ambient temperature as an exogenous input.}
The ambient temperature \(T_a(t)\) is treated as an external forcing. In typical usage scenarios, \(T_a\) varies slowly compared to the electrical dynamics.
\end{enumerate}
\subsection{Aging Assumptions}\label{sec:assumptions_aging}
\begin{enumerate}
\item \textbf{Slow-time-scale degradation.}
The state-of-health \(S(t)\) evolves on a slower time scale than \(z(t)\), \(v_p(t)\), and \(T_b(t)\). Over short horizons (single discharge), \(S\) may be approximated as quasi-static; over longer horizons, cumulative degradation is captured by the aging ODE.
\item \textbf{SEI-dominated capacity fade surrogate.}
Capacity fade is represented by a compact SEI-driven rate law:
\[
\dot S=-\lambda_{\mathrm{sei}}|I|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right),
\qquad 0\le m\le 1,
\]
which captures acceleration with higher current magnitude and higher temperature. More detailed mechanistic extensions (e.g., explicit SEI thickness) are outside the present scope.
\item \textbf{Aging impacts through resistance and effective capacity.}
The influence of \(S\) on instantaneous discharge behavior is mediated through (i) an SOH correction in the ohmic resistance \(R_0(T_b,S)\) and (ii) a proportional scaling of effective capacity \(Q_{\mathrm{eff}}(T_b,S)\). Other aging pathways (e.g., lithium plating, impedance spectra changes beyond a single RC branch) are neglected.
\end{enumerate}
\subsection{Boundaries and Applicability}\label{sec:assumptions_scope}
The proposed model is intended for \emph{discharge-dominated} smartphone operation under typical environmental conditions and is not designed to capture the following regimes without further extensions:
\begin{enumerate}
\item \textbf{Fast charging or charging--discharging transients.}
Charging dynamics, CC--CV charging protocols, and charger-induced thermal effects are not modeled.
\item \textbf{Extreme temperatures and protection-layer behavior.}
Very low-temperature operation (where diffusion limitations, severe capacity loss, or protection circuitry dominates) and very high temperatures beyond normal thermal management limits are outside scope.
\item \textbf{Severe voltage-sag and hardware protection events.}
Hard cutoffs triggered by hardware protection (e.g., overcurrent, undervoltage lockout, or thermal shutdown) are approximated by the terminal-voltage cutoff \(V_{\mathrm{cut}}\) and the CPL feasibility indicator; detailed PMIC internal logic is not explicitly modeled.
\item \textbf{Detailed multi-physics and spatial effects.}
Spatially distributed thermal fields, electrode-level electrochemical PDE models, and multi-cell balancing are not included; the goal is a compact mechanism-driven model suitable for scenario simulation and sensitivity analysis.
\item \textbf{Application-specific internal scheduling.}
Fine-grained OS scheduling, DVFS at sub-second resolution, and app-level state machines are abstracted into the exogenous inputs \(L(t),C(t),N(t),\Psi(t)\) and (optionally) the derating function \(I_{\max}(T_b)\).
\end{enumerate}
In summary, these assumptions yield a compact continuous-time model that remains physically interpretable, numerically stable, and sufficiently expressive to study runtime prediction, voltage-collapse risk, and the impact of temperature and aging under representative smartphone usage patterns.

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\section{Model Formulation}\label{sec:model}
We develop a mechanism-driven continuous-time model for smartphone battery drain that couples
(i) component-level power mapping from user/device inputs,
(ii) an equivalent-circuit battery model (ECM) with polarization memory,
(iii) a constant-power-load (CPL) algebraic closure for the discharge current,
(iv) lumped thermal dynamics, and
(v) slow health degradation (SOH).
All symbols are used consistently throughout.
\subsection{Total Power Decomposition $P_{\rm tot}$ (Screen/CPU/Network)}\label{sec:ptot}
Let the state vector be
\begin{equation}
\mathbf{x}(t)=\big[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)\big]^\top,
\end{equation}
where $z$ is the state-of-charge (SOC), $v_p$ is the polarization voltage, $T_b$ is the battery temperature,
$S$ is the state-of-health (SOH, capacity fraction), and $w$ is a continuous radio-tail state.
The exogenous input vector is
\begin{equation}
\mathbf{u}(t)=\big[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)\big]^\top,
\end{equation}
where $L$ is screen brightness, $C$ is CPU load, $N$ is network activity intensity,
$\Psi$ is signal quality (larger is better), and $T_a$ is ambient temperature.
We model the instantaneous total power demand as an additive decomposition
\begin{equation}\label{eq:ptot_def}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)),
\end{equation}
where $P_{\mathrm{bg}}$ is background/baseline power. The component mappings are chosen to be explicit and
mechanism-consistent:
\begin{align}
P_{\mathrm{scr}}(L)&=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1,\label{eq:pscr}\\
P_{\mathrm{cpu}}(C)&=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1,\label{eq:pcpu}\\
P_{\mathrm{net}}(N,\Psi,w)&=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,
\qquad \kappa>0,\ \varepsilon>0.\label{eq:pnet}
\end{align}
Here $(\Psi+\varepsilon)^{-\kappa}$ captures the increased radio power required under poor signal quality,
and $k_{\mathrm{tail}}w$ represents residual ``tail'' consumption after network bursts.
\subsection{Continuous Radio-Tail Dynamics $w(t)$}\label{sec:tail}
Instead of a discrete finite-state-machine tail model, we introduce a continuous tail state $w(t)\in[0,1]$:
\begin{equation}\label{eq:w_dyn}
\dot w(t)=\frac{\sigma(N(t))-w(t)}{\tau(N(t))},
\end{equation}
where
\begin{equation}\label{eq:sigma_tau}
\sigma(N)=\min(1,N),\qquad
\tau(N)=
\begin{cases}
\tau_\uparrow, & \sigma(N)\ge w,\\
\tau_\downarrow,& \sigma(N)< w,
\end{cases}
\qquad \tau_\uparrow\ll\tau_\downarrow.
\end{equation}
This formulation yields fast engagement of the tail state during activity increases and slow decay after activity
drops, while maintaining continuity and numerical robustness.
\subsection{ECM Terminal Voltage Equation}\label{sec:ecm}
We adopt a first-order Thevenin ECM with an ohmic resistance and one polarization branch:
\begin{equation}\label{eq:vterm}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z(t)) - v_p(t) - I(t)\,R_0(T_b(t),S(t)),
\end{equation}
where $V_{\mathrm{oc}}(z)$ is the open-circuit voltage (OCV) as a function of SOC, and
$R_0(T_b,S)$ is the temperature- and SOH-dependent ohmic resistance.
\subsection{CPL Closure: Quadratic Current and Discriminant $\Delta$}\label{sec:cpl}
Smartphone loads are well-approximated as constant-power over short time scales.
We therefore impose a CPL constraint:
\begin{equation}\label{eq:cpl}
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t)
=\big(V_{\mathrm{oc}}(z)-v_p-I R_0(T_b,S)\big)I.
\end{equation}
This yields a quadratic equation in $I$ with discriminant
\begin{equation}\label{eq:delta}
\Delta(t)=\big(V_{\mathrm{oc}}(z)-v_p\big)^2-4R_0(T_b,S)P_{\mathrm{tot}}(t).
\end{equation}
Feasibility requires $\Delta(t)\ge 0$. When feasible, the physically consistent branch is
\begin{equation}\label{eq:I_cpl}
I_{\mathrm{CPL}}(t)=\frac{V_{\mathrm{oc}}(z)-v_p-\sqrt{\Delta(t)}}{2R_0(T_b,S)}.
\end{equation}
If $\Delta(t)<0$, the demanded power is not deliverable under the CPL assumption, indicating voltage-collapse risk.
\subsection{Coupled ODEs: SOC--Polarization--Thermal--SOH--Tail}\label{sec:odes}
Given $I(t)$, the coupled state dynamics are
\begin{align}
\dot z(t)&=-\frac{I(t)}{3600\,Q_{\mathrm{eff}}(T_b(t),S(t))},\label{eq:dz}\\
\dot v_p(t)&=\frac{I(t)}{C_1}-\frac{v_p(t)}{R_1C_1},\label{eq:dvp}\\
\dot T_b(t)&=\frac{1}{C_{\mathrm{th}}}\Big(I(t)^2R_0(T_b,S)+\frac{v_p(t)^2}{R_1}-hA\big(T_b(t)-T_a(t)\big)\Big),\label{eq:dTb}\\
\dot S(t)&=-\lambda_{\mathrm{sei}}|I(t)|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b(t)}\right),\qquad 0\le m\le 1,\label{eq:dS}\\
\dot w(t)&=\frac{\sigma(N(t))-w(t)}{\tau(N(t))}.\label{eq:dw}
\end{align}
Equation \eqref{eq:dTb} uses a nonnegative polarization dissipation term $v_p^2/R_1$ for energetic consistency.
\subsection{Constitutive Relations: OCV, $R_0(T_b,S)$, and $Q_{\rm eff}(T_b,S)$}\label{sec:constitutive}
\paragraph{OCV (modified Shepherd).}
We use a modified Shepherd form:
\begin{equation}\label{eq:voc_raw}
V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z}-1\Big)+A e^{-B(1-z)}.
\end{equation}
\paragraph{Ohmic resistance with Arrhenius temperature dependence and SOH correction.}
\begin{equation}\label{eq:R0}
R_0(T_b,S)=R_{\mathrm{ref}}\exp\!\Big[\frac{E_a}{R_g}\Big(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\Big)\Big]\big(1+\eta_R(1-S)\big).
\end{equation}
\paragraph{Effective capacity.}
\begin{equation}\label{eq:Qeff}
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}\,S\Big[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\Big]_+,
\qquad [x]_+=\max(x,0).
\end{equation}
\subsection{Incorporating Three Lightweight Refinements}\label{sec:refinements}
To improve robustness while preserving the mechanistic structure, we incorporate three ``micro-refinements.''
\paragraph{(i) Low-SOC singularity protection in $V_{\mathrm{oc}}$.}
The term $1/z$ in \eqref{eq:voc_raw} is numerically singular as $z\to 0$.
We introduce an effective SOC
\begin{equation}\label{eq:zeff}
z_{\mathrm{eff}}(t)=\max\{z(t),z_{\min}\},
\end{equation}
with a small reserve threshold $z_{\min}\in(0,1)$ (e.g., $z_{\min}=0.02$) reflecting a practical BMS ``unavailable''
low-SOC region. We then evaluate OCV using $z_{\mathrm{eff}}$:
\begin{equation}\label{eq:voc}
V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z_{\mathrm{eff}}}-1\Big)+A e^{-B(1-z_{\mathrm{eff}})}.
\end{equation}
\paragraph{(ii) Nonnegative polarization heating.}
Thermal generation is written as $I^2R_0+v_p^2/R_1$, which is always nonnegative and aligns with resistive dissipation
in the polarization branch. This choice avoids sign ambiguities that can arise with alternative $Iv_p$ forms.
\paragraph{(iii) Lightweight current saturation (throttling/PMIC limiting).}
Real devices may throttle performance or limit current under low voltage or high temperature.
We model this with a temperature-dependent current cap:
\begin{equation}\label{eq:I_sat}
I(t)=\min\big(I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\big),
\end{equation}
where a simple continuous form is
\begin{equation}\label{eq:Imax}
I_{\max}(T_b)=I_{\max,0}\Big[1-\rho_T\,(T_b-T_{\mathrm{ref}})\Big]_+,\qquad \rho_T\ge 0.
\end{equation}
When $I_{\mathrm{CPL}}>I_{\max}$, the device operates in a degraded regime with delivered power
$P_{\mathrm{del}}(t)=V_{\mathrm{term}}(t)I(t)\le P_{\mathrm{tot}}(t)$, corresponding to throttling.
\subsection{Initial Conditions and Termination Definitions (TTE and optional $t_\Delta$)}\label{sec:ic_tte}
We use
\begin{equation}\label{eq:ic}
z(0)=z_0,\qquad v_p(0)=0,\qquad T_b(0)=T_a(0),\qquad S(0)=S_0,\qquad w(0)=0.
\end{equation}
We define the time-to-end (time-to-empty / time-to-shutdown) as
\begin{equation}\label{eq:TTE}
\mathrm{TTE}=\inf\Big\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \ \text{or}\ \ z(t)\le 0\Big\}.
\end{equation}
Optionally, to quantify CPL infeasibility as a voltage-collapse risk indicator, we define
\begin{equation}\label{eq:tDelta}
t_{\Delta}=\inf\Big\{t>0:\ \Delta(t)\le 0\Big\}.
\end{equation}
With throttling \eqref{eq:I_sat}, $t_\Delta$ is interpreted as the onset time at which pure CPL operation becomes
infeasible, even if the system may continue operating in a degraded mode.
\subsection{Closed-Loop Structure Summary}\label{sec:summary_loop}
The model forms a closed-loop chain:
\begin{equation}\label{eq:loop}
\mathbf{u}(t)\ \Rightarrow\ P_{\mathrm{tot}}(t)\ \Rightarrow\
\big(V_{\mathrm{oc}}(z_{\mathrm{eff}}),R_0(T_b,S),\Delta(t)\big)\ \Rightarrow\
I(t)\ \Rightarrow\ \dot{\mathbf{x}}(t)\ \Rightarrow\ \big(V_{\mathrm{term}}(t),z(t),\mathrm{TTE}\big).
\end{equation}
Nonlinear feedback arises because $P_{\mathrm{tot}}$ is enforced via CPL, while $R_0$ and $Q_{\mathrm{eff}}$
depend on $(T_b,S)$, which in turn evolve under the dissipated power.
\subsection{(Optional) Scaling and Time-Scale Discussion}\label{sec:scaling}
Although not required for computation, a brief scale analysis clarifies stiffness and numerical choices.
Let $\tau_p=R_1C_1$ denote the polarization time constant, and $\tau_{\mathrm{th}}=C_{\mathrm{th}}/(hA)$ the thermal
time constant. Typically $\tau_p\ll \tau_{\mathrm{th}}$, implying fast electrical transients and slower thermal drift.
Moreover, the tail dynamics introduce $\tau_\uparrow\ll \tau_\downarrow$.
These separated time scales motivate a time step that resolves $\tau_p$ and $\tau_\uparrow$ in explicit integration,
as enforced later in the numerical method.

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% =========================
% Section 6: Numerical Solution & Identification
% =========================
\section{Numerical Solution and Parameter Identification}
\label{sec:numerics_id}
This section describes a reproducible computational implementation of the coupled
ODE--algebraic closure induced by the constant-power-load (CPL) constraint, and a
mechanism-driven parameter identification workflow. We employ an explicit fourth-order
Runge--Kutta integrator (RK4) with a nested algebraic evaluation of the discharge current
at each substage, adaptive step-halving for convergence control, and event detection for
the time-to-end (TTE). Parameter estimation is performed by targeted sub-experiments and
log-based regressions that preserve physical interpretability.
\subsection{RK4 with substage nested algebraic evaluation of $I$}
\label{subsec:rk4_nestedI}
The state vector is $\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top$ and the input vector is
$\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top$. For any $(\mathbf{x},\mathbf{u})$, we compute
the total power demand
\begin{equation}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)).
\label{eq:Ptot_def_sec6}
\end{equation}
The terminal voltage satisfies the ECM relation
\begin{equation}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z_{\mathrm{eff}}(t))-v_p(t)-I(t)\,R_0(T_b(t),S(t)),
\label{eq:Vterm_sec6}
\end{equation}
with the low-SOC protection $z_{\mathrm{eff}}(t)=\max\{z(t),z_{\min}\}$. Under the CPL assumption,
\begin{equation}
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t)
=\big(V_{\mathrm{oc}}(z_{\mathrm{eff}})-v_p-I R_0\big)\,I,
\label{eq:CPL_sec6}
\end{equation}
which yields the discriminant
\begin{equation}
\Delta(t)=\big(V_{\mathrm{oc}}(z_{\mathrm{eff}})-v_p\big)^2-4R_0(T_b,S)\,P_{\mathrm{tot}}(t).
\label{eq:Delta_sec6}
\end{equation}
If $\Delta(t)\ge 0$, the physically consistent branch of the quadratic solution is
\begin{equation}
I_{\mathrm{CPL}}(t)=\frac{V_{\mathrm{oc}}(z_{\mathrm{eff}}(t))-v_p(t)-\sqrt{\Delta(t)}}{2R_0(T_b(t),S(t))}.
\label{eq:Icpl_sec6}
\end{equation}
To reflect device-side protection (PMIC current limiting / OS throttling), we apply a
temperature-dependent saturation
\begin{equation}
I(t)=\min\big(I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\big),
\qquad
I_{\max}(T_b)=I_{\max,0}\big[1-\rho_T\,(T_b-T_{\mathrm{ref}})\big]_+.
\label{eq:I_limit_sec6}
\end{equation}
When $\Delta(t)<0$, CPL delivery is infeasible. In such cases we record a ``collapse-risk''
event (see Section~\ref{subsec:event_detection}) and place the system in a strong
degradation regime by taking $I(t)=I_{\max}(T_b(t))$ (equivalently, one may cap
$P_{\mathrm{tot}}$), while the runtime termination (TTE) is still defined by voltage/SOC cutoffs.
Given the current mapping $I=I(\mathbf{x},\mathbf{u})$, the ODE right-hand side is evaluated using
the established dynamics:
\begin{align}
\dot z &= -\frac{I}{3600\,Q_{\mathrm{eff}}(T_b,S)}, \label{eq:zdot_sec6}\\
\dot v_p &= \frac{I}{C_1}-\frac{v_p}{R_1C_1}, \label{eq:vpdot_sec6}\\
\dot T_b &= \frac{1}{C_{\mathrm{th}}}\Big(I^2R_0(T_b,S)+\frac{v_p^2}{R_1}-hA\,(T_b-T_a)\Big), \label{eq:Tbdot_sec6}\\
\dot S &= -\lambda_{\mathrm{sei}}|I|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right), \label{eq:Sdot_sec6}\\
\dot w &= \frac{\sigma(N)-w}{\tau(N)}, \quad \sigma(N)=\min(1,N), \quad
\tau(N)=\begin{cases}\tau_\uparrow,&\sigma(N)\ge w,\\ \tau_\downarrow,&\sigma(N)<w.\end{cases}
\label{eq:wdot_sec6}
\end{align}
We discretize in time with RK4:
\begin{equation}
\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),
\label{eq:rk4_update_sec6}
\end{equation}
where each stage $\mathbf{k}_j=f(\mathbf{x},\mathbf{u},I(\mathbf{x},\mathbf{u}))$ is computed at
the corresponding intermediate state and time, and \emph{each} stage uses the nested algebraic
evaluation \eqref{eq:Delta_sec6}--\eqref{eq:I_limit_sec6}. Inputs $\mathbf{u}(t)$ at intermediate
times are obtained by interpolation (piecewise-constant or piecewise-linear) or by direct
evaluation if $\mathbf{u}(t)$ is generated procedurally.
\subsection{Projection, physical constraints, and robustness}
\label{subsec:projection_robustness}
To prevent numerical drift outside admissible ranges, we apply a mild projection after each
successful time step:
\begin{equation}
z\leftarrow \min(1,\max(0,z)),\quad
S\leftarrow \min(1,\max(0,S)),\quad
w\leftarrow \min(1,\max(0,w)).
\label{eq:projection_sec6}
\end{equation}
This projection is used only to suppress floating-point accumulation and does not change the
continuous model definition. Additionally, the low-SOC protection is enforced solely in the OCV
calculation via $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$ to avoid the $(1/z)$ singularity while preserving
the runtime termination criterion based on $z(t)$ and $V_{\mathrm{term}}(t)$.
Robustness safeguards include: (i) enforcing $Q_{\mathrm{eff}}(T_b,S)\ge 0$ via the $[\cdot]_+$ operator;
(ii) recording infeasibility when $\Delta<0$; and (iii) saturating current by
\eqref{eq:I_limit_sec6}, which prevents unrealistically large currents under low-voltage conditions.
\subsection{Step-size selection, stability, and convergence control}
\label{subsec:stepsize_convergence}
The fastest electrical time scale is the polarization time constant
\begin{equation}
\tau_p=R_1C_1.
\label{eq:taup_sec6}
\end{equation}
To resolve the polarization dynamics and avoid stage-level oscillations in an explicit method,
we impose the time-step bound
\begin{equation}
\Delta t \le 0.05\,\tau_p.
\label{eq:dt_bound_sec6}
\end{equation}
When tail dynamics are active, one may further restrict $\Delta t \le 0.05\,\tau_\uparrow$.
We enforce convergence through step-halving. Over a candidate step from $t_n$ to $t_{n+1}=t_n+\Delta t$,
we compute two solutions: one using a single RK4 step of size $\Delta t$ and another using two RK4
steps of size $\Delta t/2$. The step is accepted if
\begin{equation}
\left\|z_{\Delta t}-z_{\Delta t/2}\right\|_\infty < 10^{-4},
\label{eq:step_halving_soc_sec6}
\end{equation}
and, for runtime outputs, the inferred TTE changes by less than $1\%$ under step-halving in the
same scenario:
\begin{equation}
\frac{\big|\mathrm{TTE}_{\Delta t}-\mathrm{TTE}_{\Delta t/2}\big|}{\mathrm{TTE}_{\Delta t/2}} < 1\%.
\label{eq:step_halving_tte_sec6}
\end{equation}
If the criterion fails, we set $\Delta t\leftarrow \Delta t/2$ and recompute the step.
\subsection{Event detection and TTE interpolation}
\label{subsec:event_detection}
The runtime termination time is defined as
\begin{equation}
\mathrm{TTE}=\inf\left\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\right\}.
\label{eq:TTE_def_sec6}
\end{equation}
During integration, we monitor $V_{\mathrm{term}}(t)-V_{\mathrm{cut}}$ and $z(t)$ for sign changes. If
$V_{\mathrm{term}}(t_n)>V_{\mathrm{cut}}$ but $V_{\mathrm{term}}(t_{n+1})\le V_{\mathrm{cut}}$, we approximate
the crossing time by linear interpolation:
\begin{equation}
t_\star \approx t_n + \Delta t\,\frac{V_{\mathrm{term}}(t_n)-V_{\mathrm{cut}}}{V_{\mathrm{term}}(t_n)-V_{\mathrm{term}}(t_{n+1})}.
\label{eq:interp_voltage_event_sec6}
\end{equation}
Similarly, if $z(t_n)>0$ and $z(t_{n+1})\le 0$,
\begin{equation}
t_\star \approx t_n + \Delta t\,\frac{z(t_n)}{z(t_n)-z(t_{n+1})}.
\label{eq:interp_soc_event_sec6}
\end{equation}
If both events occur within the same step, we take the earlier of the two interpolated times as TTE.
In addition, we optionally record a CPL infeasibility (voltage-collapse risk) time
\begin{equation}
t_\Delta=\inf\{t>0:\ \Delta(t)\le 0\},
\label{eq:tDelta_def_sec6}
\end{equation}
which is useful for diagnosing ``sudden shutdown'' risk even when current limiting postpones the
actual cutoff event.
\subsection{Algorithm 3: Simulation procedure}
\label{subsec:algorithm3}
\begin{algorithm}[t]
\caption{RK4 simulation with nested CPL current evaluation and event handling}
\label{alg:rk4_cpl}
\begin{algorithmic}[1]
\REQUIRE Initial state $\mathbf{x}(0)=[z_0,0,T_a(0),S_0,0]^\top$, input trajectory $\mathbf{u}(t)$,
parameters $\Theta$, cutoff $V_{\mathrm{cut}}$, step bound $\Delta t_{\max}$.
\ENSURE Trajectories $\mathbf{x}(t)$, $V_{\mathrm{term}}(t)$, and $\mathrm{TTE}$ (and optionally $t_\Delta$).
\STATE Set $t\leftarrow 0$, $\mathbf{x}\leftarrow \mathbf{x}(0)$, choose $\Delta t\le \Delta t_{\max}$.
\STATE Initialize flags: $\texttt{risk\_recorded}\leftarrow \texttt{false}$.
\WHILE{$V_{\mathrm{term}}(t)>V_{\mathrm{cut}}$ \AND $z(t)>0$}
\STATE Evaluate $\mathbf{u}(t)$ and (if needed) $\mathbf{u}(t+\Delta t/2)$, $\mathbf{u}(t+\Delta t)$.
\STATE Compute $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$, then $V_{\mathrm{oc}}(z_{\mathrm{eff}})$, $R_0(T_b,S)$,
$Q_{\mathrm{eff}}(T_b,S)$, and $P_{\mathrm{tot}}$ via \eqref{eq:Ptot_def_sec6}.
\STATE Compute $\Delta$ via \eqref{eq:Delta_sec6}.
\IF{$\Delta<0$}
\IF{\NOT $\texttt{risk\_recorded}$}
\STATE Record $t_\Delta\leftarrow t$; $\texttt{risk\_recorded}\leftarrow \texttt{true}$.
\ENDIF
\STATE Set $I\leftarrow I_{\max}(T_b)$ \COMMENT{strong degradation / protection}
\ELSE
\STATE Compute $I_{\mathrm{CPL}}$ via \eqref{eq:Icpl_sec6} and apply saturation \eqref{eq:I_limit_sec6}.
\ENDIF
\STATE Perform one RK4 step \eqref{eq:rk4_update_sec6} with nested current evaluation at each substage.
\STATE Apply projection \eqref{eq:projection_sec6}.
\STATE Step-halving check: compare $\Delta t$ vs.\ two half-steps; if \eqref{eq:step_halving_soc_sec6} fails, set
$\Delta t\leftarrow \Delta t/2$ and recompute this step.
\STATE Update $V_{\mathrm{term}}$ via \eqref{eq:Vterm_sec6}; test event conditions.
\IF{event detected within this step}
\STATE Interpolate event time using \eqref{eq:interp_voltage_event_sec6} or \eqref{eq:interp_soc_event_sec6}.
\STATE Set $\mathrm{TTE}\leftarrow t_\star$ and \textbf{break}.
\ENDIF
\STATE Update $t\leftarrow t+\Delta t$.
\ENDWHILE
\RETURN $\mathrm{TTE}$, trajectories, and optionally $t_\Delta$.
\end{algorithmic}
\end{algorithm}
\subsection{Overall strategy for parameter identification}
\label{subsec:id_strategy}
Parameters are grouped to enable targeted identification with minimal confounding:
\begin{itemize}
\item \textbf{Open-circuit voltage (OCV)} parameters $(E_0,K,A,B)$ are identified from an OCV--SOC curve.
\item \textbf{ECM electrical parameters} $(R_0,R_1,C_1)$ are identified from current pulse tests by separating the
instantaneous ohmic drop from the relaxation dynamics.
\item \textbf{Thermal parameters} $(C_{\mathrm{th}},hA)$ are identified from heating and cooling transients.
\item \textbf{Aging parameters} $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$ are identified from capacity fade data
under controlled current/temperature conditions.
\item \textbf{Device power-mapping parameters} (screen/CPU/network) are identified from controlled workload logs by
isolating each subsystem and fitting the prescribed mechanistic forms.
\item \textbf{Tail parameters} $(k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow)$ are identified from network-burst
experiments by fitting the post-burst decay shape.
\end{itemize}
This staged approach preserves physical interpretability and avoids black-box regressions.
\subsection{OCV fitting: $(E_0,K,A,B)$}
\label{subsec:ocv_fit}
Given OCV--SOC samples $\{(z_i,V_i)\}_{i=1}^M$ collected under quasi-equilibrium conditions, we estimate
$(E_0,K,A,B)$ by least squares using the protected SOC $z_{i,\mathrm{eff}}=\max\{z_i,z_{\min}\}$:
\begin{equation}
\min_{E_0,K,A,B}\ \sum_{i=1}^M\left[
V_i-\left(E_0-K\left(\frac{1}{z_{i,\mathrm{eff}}}-1\right)+A e^{-B(1-z_{i,\mathrm{eff}})}\right)
\right]^2.
\label{eq:ocv_ls_sec6}
\end{equation}
The resulting OCV model is then used in the time-domain simulations through \eqref{eq:Vterm_sec6}.
\subsection{Pulse-based identification: $(R_0,R_1,C_1)$}
\label{subsec:pulse_id}
\paragraph{Ohmic resistance $R_0$.}
At fixed SOC and temperature, apply a current step of magnitude $\Delta I$ and measure the instantaneous voltage
drop $\Delta V(0^+)$, yielding
\begin{equation}
R_0 \approx \frac{\Delta V(0^+)}{\Delta I}.
\label{eq:R0_pulse_sec6}
\end{equation}
\paragraph{Polarization branch $(R_1,C_1)$.}
After removing the ohmic drop, the remaining relaxation is approximately first-order with time constant
$\tau_p=R_1C_1$. Denote the relaxation component by
$V_{\mathrm{rel}}(t)=V_{\mathrm{term}}(t)-\big(V_{\mathrm{oc}}-\Delta I\,R_0\big)$. Then
\begin{equation}
V_{\mathrm{rel}}(t)\approx -\Delta I\,R_1\,e^{-t/\tau_p},
\label{eq:relax_exp_sec6}
\end{equation}
so that a linear fit of $\ln|V_{\mathrm{rel}}(t)|$ versus $t$ yields $\tau_p$ and $R_1$, and hence
\begin{equation}
C_1=\frac{\tau_p}{R_1}.
\label{eq:C1_from_tau_sec6}
\end{equation}
\subsection{Temperature/aging coupling: $(R_{\mathrm{ref}},E_a,\eta_R,Q_{\mathrm{nom}},\alpha_Q)$}
\label{subsec:temp_aging_coupling}
\paragraph{Arrhenius temperature dependence for $R_0$.}
Measure $R_0$ at multiple temperatures $T_b^{(j)}$ (e.g., by \eqref{eq:R0_pulse_sec6}) and fit
\begin{equation}
\ln R_0^{(j)}=\ln R_{\mathrm{ref}}+\frac{E_a}{R_g}\left(\frac{1}{T_b^{(j)}}-\frac{1}{T_{\mathrm{ref}}}\right),
\label{eq:arrhenius_fit_sec6}
\end{equation}
to obtain $R_{\mathrm{ref}}$ and $E_a$.
\paragraph{SOH correction for resistance.}
Using measurements across different SOH levels $S$, fit
\begin{equation}
\frac{R_0(T_b,S)}{R_0(T_b,1)}\approx 1+\eta_R(1-S)
\label{eq:etaR_fit_sec6}
\end{equation}
to obtain $\eta_R$.
\paragraph{Effective capacity parameters.}
From capacity tests across temperatures, estimate $Q_{\mathrm{nom}}$ and $\alpha_Q$ using
\begin{equation}
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}\,S\left[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\right]_+.
\label{eq:Qeff_fit_sec6}
\end{equation}
\subsection{Power mapping identification: $(k_L,\gamma,k_C,\eta,k_N,\kappa,\ldots)$}
\label{subsec:power_mapping_id}
\paragraph{Screen mapping.}
Under controlled conditions with minimal CPU/network activity, vary brightness $L$ and measure total power.
After subtracting background and CPU baseline, fit
\begin{equation}
P_{\mathrm{scr}}(L)=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1.
\label{eq:screen_fit_sec6}
\end{equation}
\paragraph{CPU mapping.}
With fixed brightness and network conditions, apply controlled workloads to vary CPU load $C$ and fit
\begin{equation}
P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1.
\label{eq:cpu_fit_sec6}
\end{equation}
\paragraph{Network mapping and signal-quality penalty.}
At fixed throughput proxy $N=N_0$, vary signal quality $\Psi$ and fit
\begin{equation}
P_{\mathrm{net}}(N_0,\Psi,w)\approx P_{\mathrm{net},0}+k_N\frac{N_0}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w.
\label{eq:net_fit_sec6}
\end{equation}
For steady experiments where $w$ is constant or negligible, define
$\Delta P_{\mathrm{net}}(\Psi)=P_{\mathrm{net}}-P_{\mathrm{net},0}$ and fit in log space:
\begin{equation}
\ln \Delta P_{\mathrm{net}}(\Psi)\approx \ln(k_N N_0)-\kappa\ln(\Psi+\varepsilon),
\label{eq:kappa_fit_sec6}
\end{equation}
yielding $\kappa$ (slope) and $k_N$ (intercept).
\subsection{Tail parameter identification: $(k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow)$}
\label{subsec:tail_id}
Conduct a network-burst experiment: drive $N(t)$ high for a short period and then reduce it rapidly. After the burst,
$N\approx 0$ and the tail state decays approximately exponentially with time constant $\tau_\downarrow$:
\begin{equation}
w(t)\approx w(t_0)\,e^{-(t-t_0)/\tau_\downarrow},\qquad
P_{\mathrm{tail}}(t)=k_{\mathrm{tail}}w(t).
\label{eq:tail_decay_sec6}
\end{equation}
A linear fit of $\ln P_{\mathrm{tail}}(t)$ versus $t$ yields $\tau_\downarrow$, and the amplitude identifies
$k_{\mathrm{tail}}$. The rise time $\tau_\uparrow$ is obtained by fitting the initial ramp-up segment during burst onset,
consistent with $\tau_\uparrow\ll\tau_\downarrow$.
\subsection{Thermal parameter identification: $(C_{\mathrm{th}},hA)$}
\label{subsec:thermal_id}
Using a heating--cooling experiment, identify the lumped thermal time constant. During the cooling phase where
$I\approx 0$ and $v_p\approx 0$, \eqref{eq:Tbdot_sec6} reduces to
\begin{equation}
\dot T_b \approx -\frac{hA}{C_{\mathrm{th}}}(T_b-T_a),
\label{eq:cooling_sec6}
\end{equation}
so that
\begin{equation}
T_b(t)-T_a \approx (T_b(t_0)-T_a)\,e^{-(hA/C_{\mathrm{th}})(t-t_0)}.
\label{eq:cooling_exp_sec6}
\end{equation}
Fitting the exponential decay yields $hA/C_{\mathrm{th}}$. Then, using the heating phase with known heat generation
$\dot Q \approx I^2R_0+v_p^2/R_1$, one can separate $C_{\mathrm{th}}$ and $hA$.
\subsection{Aging parameter identification: $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$}
\label{subsec:aging_id}
From controlled aging data providing $S(t)$ under known $(I,T_b)$ conditions, the SEI-driven degradation model
\eqref{eq:Sdot_sec6} can be identified by log-linear regression. Approximating $\dot S$ via finite differences,
\begin{equation}
\ln(-\dot S)\approx \ln \lambda_{\mathrm{sei}} + m\ln|I| - \frac{E_{\mathrm{sei}}}{R_g}\frac{1}{T_b}.
\label{eq:aging_loglin_sec6}
\end{equation}
A multi-condition fit across varying currents and temperatures yields $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$.
This procedure preserves the mechanistic form and avoids black-box regression.
\subsection{Parameter table: nominal values, ranges, and sources}
\label{subsec:param_table}
Table~\ref{tab:param_summary} summarizes the parameters used in simulations, including nominal values and uncertainty
ranges for sensitivity analysis. Nominal values are obtained via the identification procedures above or from
manufacturer specifications / literature when direct measurements are unavailable. Ranges should be selected to
reflect measurement uncertainty and device-to-device variability (e.g., $\pm 10\%$--$\pm 20\%$ for power-map gains,
and temperature-dependent parameters constrained by Arrhenius fits).
\begin{table}[t]
\centering
\caption{Parameter summary (to be finalized): nominal values, uncertainty ranges, and sources.}
\label{tab:param_summary}
\begin{tabular}{llll}
\hline
Category & Parameter & Nominal / Range & Source / Method \\
\hline
OCV & $E_0,K,A,B$ & (fill) / (fill) & OCV--SOC LS fit \eqref{eq:ocv_ls_sec6} \\
ECM & $R_{\mathrm{ref}},E_a$ & (fill) / (fill) & Arrhenius fit \eqref{eq:arrhenius_fit_sec6} \\
ECM & $R_1,C_1$ & (fill) / (fill) & Pulse relaxation \eqref{eq:relax_exp_sec6} \\
SOH coupling & $\eta_R$ & (fill) / (fill) & Resistance vs.\ SOH \eqref{eq:etaR_fit_sec6} \\
Capacity & $Q_{\mathrm{nom}},\alpha_Q$ & (fill) / (fill) & Capacity tests \eqref{eq:Qeff_fit_sec6} \\
Thermal & $C_{\mathrm{th}},hA$ & (fill) / (fill) & Cooling/heating fits \eqref{eq:cooling_exp_sec6} \\
Aging & $\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}}$ & (fill) / (fill) & Log-linear fit \eqref{eq:aging_loglin_sec6} \\
Screen & $P_{\mathrm{scr},0},k_L,\gamma$ & (fill) / (fill) & Screen power fit \eqref{eq:screen_fit_sec6} \\
CPU & $P_{\mathrm{cpu},0},k_C,\eta$ & (fill) / (fill) & CPU power fit \eqref{eq:cpu_fit_sec6} \\
Network & $P_{\mathrm{net},0},k_N,\kappa$ & (fill) / (fill) & Signal penalty \eqref{eq:kappa_fit_sec6} \\
Tail & $k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow$ & (fill) / (fill) & Burst/decay \eqref{eq:tail_decay_sec6} \\
Protection & $I_{\max,0},\rho_T,z_{\min}$ & (fill) / (fill) & Device policy / assumption \\
\hline
\end{tabular}
\end{table}

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%===========================================================
\section{Uncertainty Quantification and Statistical Inference}
\label{sec:uq}
%===========================================================
This section extends the deterministic continuous-time framework in
Sections~\ref{sec:model_formulation}--\ref{sec:numerics} by modeling future
usage inputs as continuous-time stochastic processes and propagating the
resulting uncertainty through the mechanistic battery model. The objective is
to obtain a \emph{distribution} of time-to-end (TTE) rather than a single-point
estimate, and to quantify the global sensitivity of TTE to key parameters via
variance-based indices. Importantly, the underlying electro-thermal-aging
dynamics and the constant-power-load (CPL) closure are unchanged; randomness
enters only through exogenous inputs and (optionally) uncertain parameters.
%-----------------------------------------------------------
\subsection{Motivation and Model Choices for Random Inputs (OU / Regime Switching)}
\label{subsec:uq_motivation}
%-----------------------------------------------------------
Smartphone usage is intrinsically uncertain beyond a short forecasting horizon:
screen brightness $L(t)$, CPU load $C(t)$, network activity $N(t)$, and signal
quality $\Psi(t)$ exhibit mean-reverting fluctuations, cross-correlations, and
occasional abrupt changes (e.g., screen-off $\to$ gaming; good $\to$ poor
coverage). A purely deterministic extrapolation of $\mathbf{u}(t)$ therefore
tends to understate variability and cannot support probabilistic statements
(e.g., ``runtime exceeds $t$ with $90\%$ probability'').
We model the future input vector
\begin{equation}
\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top
\end{equation}
as a continuous-time stochastic process, while preserving the mechanistic
mapping
$\mathbf{u}(t)\mapsto P_{\mathrm{tot}}(t)\mapsto I(t)\mapsto \dot{\mathbf{x}}(t)$.
Two choices are considered:
\begin{enumerate}
\item \textbf{Bounded multivariate OU (Option U1).}
A multivariate Ornstein--Uhlenbeck process provides mean reversion and
cross-correlation in a continuous-time setting. Smooth bounding transforms
ensure physical admissibility ($L,C,N\in[0,1]$ and $\Psi$ in a prescribed
range).
\item \textbf{Regime-switching OU (Option U2).}
A continuous-time Markov chain $r(t)$ captures discrete ``modes'' (idle,
browsing, video, gaming; good/poor coverage). Within each regime, an OU process
drives the latent inputs. This yields bursty but still continuous trajectories.
\end{enumerate}
Both options are mechanism-compatible and avoid black-box regression: the
battery physics remain deterministic conditional on the sampled input path.
%-----------------------------------------------------------
\subsection{Mathematical Definitions and Bounding Maps}
\label{subsec:uq_definitions}
%-----------------------------------------------------------
\paragraph{Option U1: Bounded multivariate OU.}
Let $\mathbf{y}(t)\in\mathbb{R}^4$ denote latent (unbounded) Gaussian processes
associated with $[L,C,N,\Psi]$. We define
\begin{equation}
d\mathbf{y}(t)=\mathbf{K}\big(\boldsymbol{\mu}-\mathbf{y}(t)\big)\,dt
+\mathbf{\Sigma}\,d\mathbf{W}(t),
\label{eq:mvou}
\end{equation}
where $\mathbf{K}\succ 0$ controls correlation times, $\boldsymbol{\mu}$ is the
long-run mean, $\mathbf{\Sigma}$ sets diffusion intensity, and $\mathbf{W}(t)$
is a standard $4$-dimensional Brownian motion. Cross-channel correlations are
encoded in $\mathbf{\Sigma}\mathbf{\Sigma}^\top$.
Ambient temperature is modeled separately as a scalar OU process:
\begin{equation}
dT_a(t)=k_a\big(\mu_a-T_a(t)\big)\,dt+\sigma_a\,dW_a(t).
\label{eq:ou_ta}
\end{equation}
To enforce physical bounds, we map latent variables to admissible inputs using
a smooth logistic transform $\sigma(s)=(1+e^{-s})^{-1}$:
\begin{align}
L(t)&=\sigma\!\big(y_L(t)\big),\qquad
C(t)=\sigma\!\big(y_C(t)\big),\qquad
N(t)=\sigma\!\big(y_N(t)\big), \label{eq:bound_lcn}\\
\Psi(t)&=\Psi_{\min}+(\Psi_{\max}-\Psi_{\min})\,\sigma\!\big(y_\Psi(t)\big).
\label{eq:bound_psi}
\end{align}
This choice yields continuous trajectories and avoids nonphysical discontinuous
jumps that could artificially trigger the CPL infeasibility condition.
\paragraph{Option U2: Regime-switching OU.}
Let $r(t)\in\{1,\dots,R\}$ be a continuous-time Markov chain with generator
matrix $\mathbf{Q}=[q_{ij}]$, where $q_{ij}\ge 0$ for $j\neq i$ and
$q_{ii}=-\sum_{j\neq i}q_{ij}$. Conditional on $r(t)$, we define
\begin{equation}
d\mathbf{y}(t)=\mathbf{K}_{r(t)}\big(\boldsymbol{\mu}_{r(t)}-\mathbf{y}(t)\big)\,dt
+\mathbf{\Sigma}_{r(t)}\,d\mathbf{W}(t),
\label{eq:rsou}
\end{equation}
and map $\mathbf{y}(t)$ to $\{L,C,N,\Psi\}$ using
Eqs.~\eqref{eq:bound_lcn}--\eqref{eq:bound_psi}. Ambient temperature can also be
regime dependent:
\begin{equation}
dT_a(t)=k_{a,r(t)}\big(\mu_{a,r(t)}-T_a(t)\big)\,dt+\sigma_{a,r(t)}\,dW_a(t).
\label{eq:rsou_ta}
\end{equation}
This formulation captures abrupt mode changes while keeping inputs continuous
between switching times.
%-----------------------------------------------------------
\subsection{Discrete-Time Input Generation (Update Equations)}
\label{subsec:uq_generation}
%-----------------------------------------------------------
For Monte Carlo simulation, we require discrete-time updates over time step
$\Delta t$. For a scalar OU process
\begin{equation}
dy=k(\mu-y)\,dt+\sigma\,dW,
\label{eq:ou_scalar}
\end{equation}
the exact (in distribution) update is
\begin{equation}
y_{n+1}=\mu+(y_n-\mu)e^{-k\Delta t}
+\sigma\sqrt{\frac{1-e^{-2k\Delta t}}{2k}}\,\xi_n,
\qquad \xi_n\sim\mathcal{N}(0,1).
\label{eq:ou_exact}
\end{equation}
For the multivariate OU \eqref{eq:mvou}, one may use the matrix-exponential form
\begin{equation}
\mathbf{y}_{n+1}=\boldsymbol{\mu}+\mathbf{A}\big(\mathbf{y}_n-\boldsymbol{\mu}\big)
+\mathbf{B}\,\boldsymbol{\xi}_n,
\qquad \boldsymbol{\xi}_n\sim\mathcal{N}(\mathbf{0},\mathbf{I}),
\label{eq:mvou_exact}
\end{equation}
where $\mathbf{A}=e^{-\mathbf{K}\Delta t}$ and $\mathbf{B}$ satisfies
$\mathbf{B}\mathbf{B}^\top=\int_0^{\Delta t}e^{-\mathbf{K}s}\mathbf{\Sigma}\mathbf{\Sigma}^\top e^{-\mathbf{K}^\top s}\,ds$.
In practice, choosing $\mathbf{K}$ diagonal yields a simple componentwise
update using \eqref{eq:ou_exact}, while correlations can be retained through
$\mathbf{\Sigma}$.
For regime switching, over sufficiently small $\Delta t$ we approximate
\begin{equation}
\mathbb{P}\big(r_{n+1}=j\,\big|\,r_n=i\big)\approx
\begin{cases}
q_{ij}\Delta t, & j\neq i,\\
1+q_{ii}\Delta t, & j=i,
\end{cases}
\label{eq:ctmc_step}
\end{equation}
then update $\mathbf{y}$ using \eqref{eq:mvou_exact} with parameters associated
with the realized regime $r_n$.
%-----------------------------------------------------------
\subsection{Monte Carlo Propagation and TTE Distribution}
\label{subsec:uq_mc}
%-----------------------------------------------------------
Let $\omega$ denote the randomness driving $\mathbf{u}(t,\omega)$ (and, if
included, uncertain parameters). For each sampled input path $\omega_m$, the
battery dynamics are integrated using the deterministic solver from
Section~\ref{sec:numerics}: RK4 with nested CPL current evaluation at each
substep, including low-SOC OCV protection $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$,
nonnegative polarization heating $v_p^2/R_1$, and the lightweight current cap
$I=\min(I_{\mathrm{CPL}},I_{\max}(T_b))$.
The runtime endpoint is defined by
\begin{equation}
\mathrm{TTE}(\omega)=\inf\left\{t>0:\; V_{\mathrm{term}}(t,\omega)\le V_{\mathrm{cut}}
\ \text{or}\ z(t,\omega)\le 0\right\}.
\label{eq:tte_uq}
\end{equation}
(Optionally, the CPL infeasibility risk time
$t_{\Delta}=\inf\{t>0:\Delta(t,\omega)\le 0\}$ may be recorded as a separate
diagnostic.)
Given $M$ independent sample paths $\{\omega_m\}_{m=1}^M$, we obtain
$\mathrm{TTE}_m=\mathrm{TTE}(\omega_m)$ and form the empirical CDF
\begin{equation}
\widehat{F}_{\mathrm{TTE}}(t)=\frac{1}{M}\sum_{m=1}^M \mathbf{1}\{\mathrm{TTE}_m\le t\}.
\label{eq:emp_cdf}
\end{equation}
The empirical mean and variance are
\begin{equation}
\widehat{\mu}_{\mathrm{TTE}}=\frac{1}{M}\sum_{m=1}^M \mathrm{TTE}_m,\qquad
\widehat{\sigma}^2_{\mathrm{TTE}}=\frac{1}{M-1}\sum_{m=1}^M(\mathrm{TTE}_m-\widehat{\mu}_{\mathrm{TTE}})^2.
\label{eq:tte_moments}
\end{equation}
\paragraph{Monte Carlo error.}
For standard Monte Carlo estimators of smooth functionals of TTE, the
statistical error decays as $O(M^{-1/2})$. We therefore increase $M$ until key
summaries (mean and selected quantiles) stabilize under doubling $M$.
%-----------------------------------------------------------
\subsection{Confidence Intervals, Quantiles, and Survival Curves}
\label{subsec:uq_inference}
%-----------------------------------------------------------
\paragraph{Confidence interval for the mean.}
By the central limit theorem, an approximate $95\%$ confidence interval for the
mean TTE is
\begin{equation}
\widehat{\mu}_{\mathrm{TTE}}\ \pm\ 1.96\,\frac{\widehat{\sigma}_{\mathrm{TTE}}}{\sqrt{M}}.
\label{eq:ci_mean}
\end{equation}
When $M$ is moderate and the distribution is skewed, a nonparametric bootstrap
over $\{\mathrm{TTE}_m\}$ can be used to obtain robust confidence bounds.
\paragraph{Quantiles.}
Let $\mathrm{TTE}_{(1)}\le \cdots \le \mathrm{TTE}_{(M)}$ denote the ordered
samples. The empirical $p$-quantile is
\begin{equation}
\widehat{q}_p=\mathrm{TTE}_{(\lceil pM\rceil)},\qquad p\in(0,1).
\label{eq:quantile}
\end{equation}
In particular, the median is $\widehat{q}_{0.5}$, and the lower-tail quantile
$\widehat{q}_{0.1}$ supports conservative ``guaranteed runtime'' statements.
\paragraph{Survival function.}
A reliability-style summary is the survival curve
\begin{equation}
\widehat{S}(t)=\mathbb{P}(\mathrm{TTE}>t)\approx 1-\widehat{F}_{\mathrm{TTE}}(t).
\label{eq:survival}
\end{equation}
This directly answers: ``what is the probability the phone remains operational
beyond time $t$?''
%-----------------------------------------------------------
\subsection{Variance-Based Global Sensitivity (Sobol Indices)}
\label{subsec:uq_sobol}
%-----------------------------------------------------------
We quantify global parameter importance via variance-based sensitivity indices
for the scalar quantity of interest (QoI)
\begin{equation}
Y=\mathrm{TTE}.
\end{equation}
Let $\boldsymbol{\xi}=(\xi_1,\dots,\xi_d)$ denote uncertain factors (e.g.,
$k_L,\gamma,k_N,\kappa,\mu_a$ and other parameters as needed), assumed
independent with prescribed prior distributions. Because usage randomness
$\omega$ also contributes variance, we recommend defining the QoI as the
\emph{conditional expectation} over usage paths:
\begin{equation}
Y(\boldsymbol{\xi})=\mathbb{E}_{\omega}\big[\mathrm{TTE}(\boldsymbol{\xi},\omega)\big],
\label{eq:qoi_condexp}
\end{equation}
which yields stable and actionable sensitivities to design/physics parameters.
In computations, \eqref{eq:qoi_condexp} is approximated by an inner Monte Carlo
average over $M_{\omega}$ usage realizations.
The first-order Sobol index of factor $\xi_i$ is defined as
\begin{equation}
S_i=\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid \xi_i]\big)}{\mathrm{Var}(Y)},
\label{eq:sobol_first}
\end{equation}
and the total-effect index is
\begin{equation}
S_{T_i}=1-\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid \boldsymbol{\xi}_{\sim i}]\big)}{\mathrm{Var}(Y)},
\label{eq:sobol_total}
\end{equation}
where $\boldsymbol{\xi}_{\sim i}$ denotes all factors except $\xi_i$. Large
$S_i$ indicates a strong main effect, while a large gap $S_{T_i}-S_i$ indicates
substantial interaction and/or nonlinearity (expected here due to CPL feedback
and electro-thermal coupling).
%-----------------------------------------------------------
\subsection{Saltelli Sampling and Estimation}
\label{subsec:uq_saltelli}
%-----------------------------------------------------------
We employ the Saltelli sampling scheme for efficient estimation of Sobol
indices. Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{N\times d}$ be two independent
sample matrices of $\boldsymbol{\xi}$. For each $i\in\{1,\dots,d\}$, construct
$\mathbf{A}^{(i)}_B$ by replacing the $i$-th column of $\mathbf{A}$ with the
$i$-th column of $\mathbf{B}$. Denote the corresponding model evaluations by
\begin{equation}
Y_A^{(n)}=Y(\mathbf{A}_n),\quad
Y_B^{(n)}=Y(\mathbf{B}_n),\quad
Y_{A^{(i)}_B}^{(n)}=Y(\mathbf{A}^{(i)}_{B,n}),
\qquad n=1,\dots,N.
\end{equation}
We estimate $\mathrm{Var}(Y)$ from the pooled samples and compute Sobol
estimators in the following commonly used form:
\begin{equation}
\widehat{S}_i=
\frac{\frac{1}{N}\sum_{n=1}^N Y_B^{(n)}\left(Y_{A^{(i)}_B}^{(n)}-Y_A^{(n)}\right)}
{\widehat{\mathrm{Var}}(Y)},
\label{eq:saltelli_first}
\end{equation}
\begin{equation}
\widehat{S}_{T_i}=
\frac{\frac{1}{2N}\sum_{n=1}^N \left(Y_A^{(n)}-Y_{A^{(i)}_B}^{(n)}\right)^2}
{\widehat{\mathrm{Var}}(Y)}.
\label{eq:saltelli_total}
\end{equation}
\paragraph{Nested averaging over usage paths.}
Each $Y(\cdot)$ above is computed as
\begin{equation}
Y(\boldsymbol{\xi})\approx \frac{1}{M_{\omega}}\sum_{m=1}^{M_{\omega}}
\mathrm{TTE}(\boldsymbol{\xi},\omega_m),
\label{eq:nested_mc}
\end{equation}
where $\{\omega_m\}$ are i.i.d.\ usage realizations generated by
Option~U1/U2. This inner average reduces the Monte Carlo noise in $Y$ so that
the outer Saltelli estimators converge reliably in $N$.
%-----------------------------------------------------------
\subsection{Optional: Variance Reduction (LHS / Quasi-Monte Carlo)}
\label{subsec:uq_varred}
%-----------------------------------------------------------
While plain Monte Carlo converges at rate $O(M^{-1/2})$, variance reduction can
improve efficiency when computational budgets are tight.
\paragraph{Latin hypercube sampling (LHS).}
For estimating the TTE distribution under uncertain inputs/parameters, LHS can
replace i.i.d.\ sampling of low-dimensional uncertain parameters
$\boldsymbol{\xi}$ to reduce estimator variance without changing the model. LHS
is especially effective when the dominant uncertainty is parameter-driven.
\paragraph{Quasi-Monte Carlo (QMC).}
For Sobol estimation (outer sampling), low-discrepancy sequences (e.g., Sobol
sequences) can improve convergence of integral estimates in moderate
dimensions. In this work, QMC can be applied to generate $\mathbf{A},\mathbf{B}$
before constructing $\mathbf{A}^{(i)}_B$. Because our QoI involves a nested
average \eqref{eq:nested_mc}, QMC primarily benefits the outer parameter
integration, while the inner usage randomness still scales as
$O(M_\omega^{-1/2})$.
\paragraph{Control variates (conceptual).}
If a simplified surrogate (e.g., the same model with fixed $T_b=T_a$ or without
aging) is available, it may serve as a control variate to reduce variance of
$\mathrm{TTE}$. We do not rely on this technique in the baseline pipeline.
%-----------------------------------------------------------
\subsection{Optional: Unified Two-Level Uncertainty (Inputs and Parameters)}
\label{subsec:uq_twolevel}
%-----------------------------------------------------------
When both usage inputs and physical/power parameters are uncertain, the full
QoI can be viewed hierarchically as
\begin{equation}
\mathrm{TTE}=\mathrm{TTE}(\boldsymbol{\xi},\omega),
\end{equation}
with $\boldsymbol{\xi}$ representing uncertain parameters (e.g., $k_L,\gamma,
k_N,\kappa,\mu_a,hA$) and $\omega$ representing stochastic input realizations
from Option~U1/U2. Two complementary summaries are useful:
\paragraph{Unconditional runtime distribution.}
The overall distribution integrates over both sources of uncertainty:
\begin{equation}
F_{\mathrm{TTE}}(t)=\mathbb{P}(\mathrm{TTE}\le t)=
\int \mathbb{P}\!\left(\mathrm{TTE}(\boldsymbol{\xi},\omega)\le t\ \big|\ \boldsymbol{\xi}\right)
\,p(\boldsymbol{\xi})\,d\boldsymbol{\xi}.
\label{eq:unconditional}
\end{equation}
This is estimated by outer sampling of $\boldsymbol{\xi}$ and inner sampling of
$\omega$.
\paragraph{Sensitivity of conditional mean runtime.}
For design guidance, sensitivities are computed for
$Y(\boldsymbol{\xi})=\mathbb{E}_{\omega}[\mathrm{TTE}(\boldsymbol{\xi},\omega)]$
as in \eqref{eq:qoi_condexp}, yielding Sobol indices that reflect how parameter
variation shifts \emph{expected} runtime under random usage.
\paragraph{Practical computation.}
A computationally efficient compromise is to (i) propagate usage uncertainty
with a large $M$ at nominal parameters to obtain $F_{\mathrm{TTE}}$, and (ii)
compute Sobol indices with moderate inner averaging $M_\omega$ and outer sample
size $N$ to rank parameter importance.
%-----------------------------------------------------------
\subsection*{Algorithmic Summary}
%-----------------------------------------------------------
For completeness, the full UQ pipeline used in subsequent sections can be
summarized as follows:
\begin{itemize}
\item Generate stochastic input paths $\mathbf{u}(t,\omega)$ using
Eqs.~\eqref{eq:mvou}--\eqref{eq:bound_psi} (Option~U1) or
Eqs.~\eqref{eq:rsou}--\eqref{eq:rsou_ta} (Option~U2), with discrete updates
given by \eqref{eq:ou_exact}--\eqref{eq:ctmc_step}.
\item For each path, solve the mechanistic battery model using RK4 with nested
CPL current evaluation (Section~\ref{sec:numerics}) and record
$\mathrm{TTE}$ from \eqref{eq:tte_uq}.
\item Construct the empirical distribution \eqref{eq:emp_cdf}, compute moments
\eqref{eq:tte_moments}, confidence intervals \eqref{eq:ci_mean}, quantiles
\eqref{eq:quantile}, and survival curve \eqref{eq:survival}.
\item For global sensitivity, evaluate $Y(\boldsymbol{\xi})$ via nested averaging
\eqref{eq:nested_mc} and estimate Sobol indices with Saltelli sampling
\eqref{eq:saltelli_first}--\eqref{eq:saltelli_total}.
\end{itemize}

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flowchart TD
A\["模型已建立\\n状态 x=\[z,v\_p,T\_b,S,w]\\n输入 u=\[L,C,N,Ψ,T\_a]\\n终止: V\_term<=V\_cut 或 z<=0"] --> B\["0. 准备:场景/数据\\n给定 u(t)、初值 x(0)、参数 Θ"]
%% ---- 主仿真(确定性/单轨迹) ----
B --> C\["1. 主仿真循环t=0→…"]
C --> C1\["1.1 计算总功耗 P\_tot(u,x)\\n(背景+屏幕+CPU+网络/尾效应)"]
C1 --> C2\["1.2 计算构成关系\\nV\_oc(z\_eff), R0(T\_b,S), Q\_eff(T\_b,S)"]
C2 --> C3{"1.3 CPL 代数闭合求 I\\nΔ>=0 ?"}
C3 -- "是" --> C4\["I\_CPL 分支"]
C3 -- "否" --> C4b\["记录 t\_Δ坍塌风险\\n进入降级/限流策略"]
C4 --> C5\["1.4 限流/降级:\\nI=min(I\_CPL, I\_max(T\_b))\\n(得到实际 I 与 P\_del)"]
C4b --> C5
C5 --> C6\["1.5 ODE RHS\\nż, ṽ\_p, Ṫ\_b, Ṡ, ŵ"]
C6 --> C7\["1.6 RK4 推进(每个 stage 都嵌套求 I\\n中间时刻 u(t) 插值/生成"]
C7 --> C8\["1.7 投影/物理约束:\\nz,S,w 截断到可行区间\\nQ\_eff>=0 等保护"]
C8 --> C9{"1.8 事件检测:\\nV\_term<=V\_cut 或 z<=0 ?"}
C9 -- "否" --> C1
C9 -- "是" --> D\["2. 输出:轨迹 + TTE\\n(可线性插值得到 t\*)"]
%% ---- 鲁棒性检验 ----
D --> R{"3. 鲁棒性检验是否通过?"}
R --> R1\["3.1 步长/收敛鲁棒性:\\nΔt vs Δt/2 的 step-halving\\n||zΔt - zΔt/2||∞ < 1e-4\\n且 TTE 相对误差 < 1%"]
R --> R2\["3.2 事件定位鲁棒性:\\n电压/ SOC 过阈值的插值稳定\\n(比较不同插值/求根策略)"]
R --> R3\["3.3 CPL/低电压鲁棒性:\\nΔ<0 记录 + I 饱和\\n避免非物理大电流/数值爆炸"]
R --> R4\["3.4 约束/奇异性鲁棒性:\\nz\_eff 保护(避免 1/z 奇异)\\n状态投影抑制漂移"]
R1 \& R2 \& R3 \& R4 --> ROK{"全部通过?"}
ROK -- "否" --> Fix\["回到修正:\\n减小 Δt / 调整容差\\n检查限流策略/保护阈值\\n必要时回到参数标定 Θ"]
Fix --> C
ROK -- "是" --> E\["4. 形成确定性结论\\n场景对比、指标归因\\n(谁最耗电/何时风险高)"]
%% ---- 敏感性 + UQ ----
E --> S\["5. 全局敏感性Sobol/Saltelli\\nY=TTE 或 E\[TTE]\\n得到主效应/总效应排序"]
S --> U\["6. 不确定性量化 UQ\\n生成随机输入路径 u(t,ω)\\n(OU 或 切换OU)\\n对每条路径重复主仿真→TTE样本"]
U --> U1\["6.1 统计推断:\\n经验CDF/均值方差/CI/分位数\\n生存曲线 P(TTE>t)"]
U1 --> G\["7. 输出与建议:\\n(1) 预测 TTE(点估计+区间)\\n(2) 风险指标 t\_Δ\\n(3) 敏感因子→策略建议"]

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标记含义保持不变:✅已完成 / 🟡部分完成 / ⬜未完成。
---
# 论文全部组成(精细版)+ 完成状态(最新版)
## 封面与前置页Front Matter
1. **题目Title** ✅(已在整合稿顶部给出)
2. **队伍信息Team Control Number 等)** 🟡(目前仍是占位符,如 1111111需要你最后替换
3. **Summary Sheet / Executive SummaryMCM 一页摘要页)** ✅(已写成可提交版,含模型/结果/建议)
4. **摘要Abstract** 🟡(当前更偏 “Summary Sheet + Summary”若你坚持 IEEE/期刊格式可再补一个 150200 词 Abstract
5. **关键词Keywords** ✅(整合稿已有)
---
## 1 引言与背景Introduction & Context
1.1 研究背景与动机(续航、复杂负载、掉电问题) ✅
1.2 建模挑战CPL 闭环、热-老化、网络尾耗) ✅
1.3 本文贡献点(机制驱动 + 数据校准 + 可行性判据 + UQ
1.4 文章结构安排Paper organization35 行) 🟡(正文结构隐含在 Q1/Q2/Q3 分节里;可再补一句“本文结构如下”更标准)
---
## 2 问题重述与建模目标Problem Restatement & Objectives
2.1 题意重述 ✅Q1/Q2/Q3 每题都有对应 “问题重述”)
2.2 输入/输出与预测任务SOC/端电压/TTE/风险诊断) ✅
2.3 评价指标与场景口径TTE、可选 (t_\Delta)、电压截止口径) ✅(已落地到结果表与图)
---
## 3 符号说明与变量定义Nomenclature
3.1 状态向量 (\mathbf{x}(t))SOC/极化/温度/SOH/尾耗) 🟡(变量在文内齐了,但尚未形成“符号表格”)
3.2 输入向量 (\mathbf{u}(t))(亮度/CPU/网络/信号/环境) 🟡(同上)
3.3 输出与派生量((V_{\rm term},P_{\rm tot},\Delta) 等) 🟡
3.4 参数集合与单位说明(参数表 + 单位 + 来源) ✅
---
## 4 模型假设Assumptions
4.1 结构性假设(单电芯等效、功耗可加) ✅
4.2 负载侧假设CPL 近似、(\Delta) 可行性判据、限流) ✅
4.3 热学假设(集中参数热模型) 🟡(有提到但仍偏简略;结果图已出温度轨迹)
4.4 老化假设(慢变量/短时忽略) 🟡(目前仿真里等价于 short-horizon 不启用 SOH 动力学)
4.5 适用范围与边界条件(不适用:快充/极寒/多电芯梯度等) 🟡(已写 Weakness/Limitations但可再条列化
---
## 5 模型建立Model Formulation
5.1 总功率分解 (P_{\rm tot})(屏幕/CPU/网络/背景) ✅(并且已用数据集拟合系数落地)
5.2 网络尾耗连续动力学 (w(t)) ✅(模型已建并在仿真中启用)
5.3 Thevenin 1-RC ECM 端电压方程 ✅
5.4 CPL 闭环电流(二次解 + 判别式 (\Delta)
5.5 耦合 ODESOC极化尾耗SOH 作为可选慢变量) 🟡(短时预测已完整;老化动态仍未参数化验证)
5.6 本构关系OCV、(R_0(T,S))、(Q_{\rm eff}(T,S)) 🟡(表达已写,但参数表/辨识仍欠缺)
5.7 三条微调(护栏、非负热、限流/降频) ✅(策略接口已写)
5.8 初值与终止定义TTE、电压截止、可选 (t_\Delta)
5.9 模型闭环结构小结(输入→功耗→电流→状态→输出) ✅
5.10(可选)无量纲化/尺度分析 ⬜(锦上添花,后置)
---
## 6 数值求解与参数辨识Numerical Solution & Identification
6.1 RK4 + 代数子步求 (I) ✅
6.2 投影/物理约束((z,w,S) 截断等) ✅
6.3 步长/稳定性/收敛性说明 🟡(文字有,但“步长对半结果图/表”仍未补)
6.4 事件检测与 TTE 插值 ✅
6.5 算法流程伪代码 ✅
6.6 参数辨识总体策略(分组辨识) 🟡
6.7 **功耗映射辨识(亮度/CPU/WLAN** ✅(已生成系数表+拟合图+文字说明)
6.8 尾耗参数辨识((\tau_\uparrow,\tau_\downarrow,k_{\rm tail}) ⬜(目前用于仿真的是可用的名义值,但不是从数据拟合出来的)
6.9 热参数辨识((C_{\rm th},hA)
6.10 老化参数辨识((\lambda_{\rm sei},m,E_{\rm sei})
6.11 **最终参数表(名义值/范围/来源)**
---
## 7 不确定性建模与统计推断Uncertainty Quantification
7.1 随机输入建模动机与选择 ✅(已在 Q3 与 UQ 段落中表达)
7.2 MC 传播得到 TTE 分布 ✅(已完成:箱线图 + CDF + 汇总表)
7.3 置信区间/分位数/生存函数(可选) 🟡(分位数已做;生存曲线可后补)
7.4 Sobol/方差分解(严格版) ⬜(目前未做 Saltelli/Sobol 正式估计)
7.5 **轻量敏感性(相关性排名)** ✅(作为 Sobol 的低成本替代结果已给出)
---
## 8 仿真设置与结果Simulation Setup & Results
8.1 基准参数与初始条件设置 🟡(仿真已跑通,但“集中参数表”还没写成正式表格)
8.2 场景定义5 个代表场景 + 输入口径) ✅(已输出 step3_scenarios.csv
8.3 输出指标((z,V_{\rm term},T_b,w,\Delta) 🟡核心三条轨迹已出w/I/功耗分量还未做图)
8.4 **确定性轨迹展示**SOC/Vterm/Temp 三张综合图已生成)
8.5 MC 轨迹束/不确定性带 🟡(目前给的是分布图(箱线/CDF“轨迹束带状图”还没画
8.6 **TTE 分布(箱线图/CDF**Step4 已出)
8.7 策略前后对比(限流/降频/亮度策略) ⬜(建议后续加 12 张“策略对比表/图”会很提分)
8.8 极端条件分析(低温/老化/弱信号) 🟡(弱信号已做;低温/老化仍未系统化)
---
## 9 模型检验与讨论Verification, Validation & Discussion
9.1 数值验证(步长对半/单调性/守恒检查) ✅
9.2 行为验证(场景排序合理性) ✅TTE 排序已经通过结果表体现)
9.3 对比验证(与日志/真实数据/文献区间) ⬜(目前还没有“外部对照”)
9.4 失效机理解释((\Delta) 风险、弱信号触发) 🟡(Δ曲线已出一张 worst-case但可增强讨论
9.5 Strengths/Weaknesses/Limitations ✅(整合稿已补齐结构化讨论)
---
## 10 策略建议与结论Recommendations & Conclusion
10.1 用户侧建议(亮度/网络/后台) ✅(已写 Recommendations
10.2 系统侧策略(弱信号模式/尾耗管理/温控降频) 🟡(建议文字有,但缺“策略仿真对比”支撑)
10.3 结论总结(模型贡献 + 关键发现 + 数据支撑) ✅
10.4 未来工作Sobol、热/老化辨识、多电芯等) ✅(已写 Limitations & future work
---
## 参考文献与附录References & Appendices
R1 参考文献 ✅(整合稿已有 References
A1 推导细节CPL 二次解、判别式等) 🟡(正文已写核心推导,但“附录化/集中整理”未做)
A2 图表索引/伪代码(可选) 🟡(算法段落已有,未做“附录版整洁伪代码块”)
A3 参数表、单位、来源与范围 ⬜(与 6.11 同一缺口)
A4 额外图(拟合、收敛、更多场景) 🟡(拟合图已有;收敛/更多场景未补)
---
# 目前已完成的“关键里程碑”(更新版)
***模型核心闭环**CPL + Thevenin 1-RC + tail 动力学)
***数据校准落地**(亮度/CPU/WLAN 映射已拟合并进入仿真)
***第8节核心结果已具备**5 场景:轨迹图 + TTE 表)
***不确定性结果已具备**MC 分布:箱线图 + CDF + 区间表)
***第9/10节“收口结构”已补齐**Verification/Strengths/Weaknesses/Conclusion/Recommendations
---

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### 0. Summary Sheet1页
* 闭环链条一句话
* 关键结果数字 + 建议
### 1. Introduction & Problem Restatement
* 目标TTE + 连续时间 + 欠压/风险
### 2. Model Overview & Assumptions很重要
* 给一张框图:(\mathbf{u}\to P_{\text{tot}}\to I\to \dot{\mathbf{x}}\to) TTE
* **先定义**(P_{\text{tot}}(t)) 是“负载对电池的等效功率需求”(细节后面再展开)
* 输出TTE + 风险时间(如 (t_\Delta)
### 3. Battery-side Model先讲电池
3.1 First-order Thevenin ECM(V_{\text{term}})
3.2 极化支路动力学
3.3 热模型(耗散项)
3.4 老化/SOH强调短时标可冻结、长时标再更新
### 4. Load-side Power Model再讲功耗来源
4.1 组件功耗分解:屏幕/CPU/网络/后台
4.2 网络弱信号惩罚 + tail 状态 (w(t))
4.3 场景表Standby/Video/Gaming/Navigation/Weak signal 等)→ 映射到 (P_{\text{tot}}(t))
### 5. Coupling & Governing Equations最后把闭环收口
5.1 CPL 闭环:由 (P_{\text{tot}}(t)=V_{\text{term}}(t)I(t)) 解 (I(t))(含判别式 (\Delta)
5.2 “(\Delta)”的物理意义(负阻抗/崩溃风险)
5.3 总耦合 ODE把电池 + tail + 热 + 老化拼成最终系统)
5.4 终止条件TTE 与 (t_\Delta) 的区分(避免逻辑矛盾)
### 6. Numerical Method & Parameter Strategy
* 求解器、步长控制、事件检测、参数分组/标定思路
### 7. Results & Insights
* 各场景曲线 + TTE
* (t_\Delta) vs TTE“突然关机风险”亮点
* 策略建议(用户/系统)
### 8. Uncertainty Quantification & Sensitivity
* OU/切换 OU、Sobol、置信区间、相对不确定度等
### 9. Discussion / Limitations / Extensions
* 2-RC 放这里或附录(主文不强上)
### References + Appendices
* 完整假设、推导、参数表、伪代码

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标记含义保持不变:✅已完成 / 🟡部分完成 / ⬜未完成。
---
# 论文全部组成(精细版)+ 完成状态(最新版)
## 封面与前置页Front Matter
1. **题目Title** ✅(已在整合稿顶部给出)
2. **队伍信息Team Control Number 等)** 🟡(目前仍是占位符,如 1111111需要你最后替换
3. **Summary Sheet / Executive SummaryMCM 一页摘要页)** ✅(已写成可提交版,含模型/结果/建议)
4. **摘要Abstract** 🟡(当前更偏 “Summary Sheet + Summary”若你坚持 IEEE/期刊格式可再补一个 150200 词 Abstract
5. **关键词Keywords** ✅(整合稿已有)
---
## 1 引言与背景Introduction & Context
1.1 研究背景与动机(续航、复杂负载、掉电问题) ✅
1.2 建模挑战CPL 闭环、热-老化、网络尾耗) ✅
1.3 本文贡献点(机制驱动 + 数据校准 + 可行性判据 + UQ
1.4 文章结构安排Paper organization35 行) 🟡(正文结构隐含在 Q1/Q2/Q3 分节里;可再补一句“本文结构如下”更标准)
---
## 2 问题重述与建模目标Problem Restatement & Objectives
2.1 题意重述 ✅Q1/Q2/Q3 每题都有对应 “问题重述”)
2.2 输入/输出与预测任务SOC/端电压/TTE/风险诊断) ✅
2.3 评价指标与场景口径TTE、可选 (t_\Delta)、电压截止口径) ✅(已落地到结果表与图)
---
## 3 符号说明与变量定义Nomenclature
3.1 状态向量 (\mathbf{x}(t))SOC/极化/温度/SOH/尾耗) 🟡(变量在文内齐了,但尚未形成“符号表格”)
3.2 输入向量 (\mathbf{u}(t))(亮度/CPU/网络/信号/环境) 🟡(同上)
3.3 输出与派生量((V_{\rm term},P_{\rm tot},\Delta) 等) 🟡
3.4 参数集合与单位说明(参数表 + 单位 + 来源) ✅
---
## 4 模型假设Assumptions
4.1 结构性假设(单电芯等效、功耗可加) ✅
4.2 负载侧假设CPL 近似、(\Delta) 可行性判据、限流) ✅
4.3 热学假设(集中参数热模型) 🟡(有提到但仍偏简略;结果图已出温度轨迹)
4.4 老化假设(慢变量/短时忽略) 🟡(目前仿真里等价于 short-horizon 不启用 SOH 动力学)
4.5 适用范围与边界条件(不适用:快充/极寒/多电芯梯度等) 🟡(已写 Weakness/Limitations但可再条列化
---
## 5 模型建立Model Formulation
5.1 总功率分解 (P_{\rm tot})(屏幕/CPU/网络/背景) ✅(并且已用数据集拟合系数落地)
5.2 网络尾耗连续动力学 (w(t)) ✅(模型已建并在仿真中启用)
5.3 Thevenin 1-RC ECM 端电压方程 ✅
5.4 CPL 闭环电流(二次解 + 判别式 (\Delta)
5.5 耦合 ODESOC极化尾耗SOH 作为可选慢变量) 🟡(短时预测已完整;老化动态仍未参数化验证)
5.6 本构关系OCV、(R_0(T,S))、(Q_{\rm eff}(T,S)) 🟡(表达已写,但参数表/辨识仍欠缺)
5.7 三条微调(护栏、非负热、限流/降频) ✅(策略接口已写)
5.8 初值与终止定义TTE、电压截止、可选 (t_\Delta)
5.9 模型闭环结构小结(输入→功耗→电流→状态→输出) ✅
5.10(可选)无量纲化/尺度分析 ⬜(锦上添花,后置)
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## 6 数值求解与参数辨识Numerical Solution & Identification
6.1 RK4 + 代数子步求 (I) ✅
6.2 投影/物理约束((z,w,S) 截断等) ✅
6.3 步长/稳定性/收敛性说明 🟡(文字有,但“步长对半结果图/表”仍未补)
6.4 事件检测与 TTE 插值 ✅
6.5 算法流程伪代码 ✅
6.6 参数辨识总体策略(分组辨识) 🟡
6.7 **功耗映射辨识(亮度/CPU/WLAN** ✅(已生成系数表+拟合图+文字说明)
6.8 尾耗参数辨识((\tau_\uparrow,\tau_\downarrow,k_{\rm tail}) ⬜(目前用于仿真的是可用的名义值,但不是从数据拟合出来的)
6.9 热参数辨识((C_{\rm th},hA)
6.10 老化参数辨识((\lambda_{\rm sei},m,E_{\rm sei})
6.11 **最终参数表(名义值/范围/来源)**
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## 7 不确定性建模与统计推断Uncertainty Quantification
7.1 随机输入建模动机与选择 ✅(已在 Q3 与 UQ 段落中表达)
7.2 MC 传播得到 TTE 分布 ✅(已完成:箱线图 + CDF + 汇总表)
7.3 置信区间/分位数/生存函数(可选) 🟡(分位数已做;生存曲线可后补)
7.4 Sobol/方差分解(严格版) ⬜(目前未做 Saltelli/Sobol 正式估计)
7.5 **轻量敏感性(相关性排名)** ✅(作为 Sobol 的低成本替代结果已给出)
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## 8 仿真设置与结果Simulation Setup & Results
8.1 基准参数与初始条件设置 🟡(仿真已跑通,但“集中参数表”还没写成正式表格)
8.2 场景定义5 个代表场景 + 输入口径) ✅(已输出 step3_scenarios.csv
8.3 输出指标((z,V_{\rm term},T_b,w,\Delta) 🟡核心三条轨迹已出w/I/功耗分量还未做图)
8.4 **确定性轨迹展示**SOC/Vterm/Temp 三张综合图已生成)
8.5 MC 轨迹束/不确定性带 🟡(目前给的是分布图(箱线/CDF“轨迹束带状图”还没画
8.6 **TTE 分布(箱线图/CDF**Step4 已出)
8.7 策略前后对比(限流/降频/亮度策略) ⬜(建议后续加 12 张“策略对比表/图”会很提分)
8.8 极端条件分析(低温/老化/弱信号) 🟡(弱信号已做;低温/老化仍未系统化)
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## 9 模型检验与讨论Verification, Validation & Discussion
9.1 数值验证(步长对半/单调性/守恒检查) ✅
9.2 行为验证(场景排序合理性) ✅TTE 排序已经通过结果表体现)
9.3 对比验证(与日志/真实数据/文献区间) ⬜(目前还没有“外部对照”)
9.4 失效机理解释((\Delta) 风险、弱信号触发) 🟡(Δ曲线已出一张 worst-case但可增强讨论
9.5 Strengths/Weaknesses/Limitations ✅(整合稿已补齐结构化讨论)
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## 10 策略建议与结论Recommendations & Conclusion
10.1 用户侧建议(亮度/网络/后台) ✅(已写 Recommendations
10.2 系统侧策略(弱信号模式/尾耗管理/温控降频) 🟡(建议文字有,但缺“策略仿真对比”支撑)
10.3 结论总结(模型贡献 + 关键发现 + 数据支撑) ✅
10.4 未来工作Sobol、热/老化辨识、多电芯等) ✅(已写 Limitations & future work
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## 参考文献与附录References & Appendices
R1 参考文献 ✅(整合稿已有 References
A1 推导细节CPL 二次解、判别式等) 🟡(正文已写核心推导,但“附录化/集中整理”未做)
A2 图表索引/伪代码(可选) 🟡(算法段落已有,未做“附录版整洁伪代码块”)
A3 参数表、单位、来源与范围 ⬜(与 6.11 同一缺口)
A4 额外图(拟合、收敛、更多场景) 🟡(拟合图已有;收敛/更多场景未补)
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# 目前已完成的“关键里程碑”(更新版)
***模型核心闭环**CPL + Thevenin 1-RC + tail 动力学)
***数据校准落地**(亮度/CPU/WLAN 映射已拟合并进入仿真)
***第8节核心结果已具备**5 场景:轨迹图 + TTE 表)
***不确定性结果已具备**MC 分布:箱线图 + CDF + 区间表)
***第9/10节“收口结构”已补齐**Verification/Strengths/Weaknesses/Conclusion/Recommendations
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