部分成文

A题/成文/1封面与前置页.md

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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 current–voltage 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 Sheet（MCM 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 Runge–Kutta 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, 5th–95th 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.
+
+---
+

A题/成文/2问题重述与建模目标.md

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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.
+%========================================================

A题/成文/3符号说明与变量定义.md

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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}.
+
+% End of Section 3

A题/成文/4模型假设.md

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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.

temp.md

@@ -0,0 +1,256 @@
+你这个问题问得很“论文落地导向”：MCM/ICM 评委最在意的不是你有没有真实手机数据，而是——**你说的参数怎么来、仿真怎么跑、结果怎么验证**。下面我按你现在的论文结构，把「哪些地方需要数据 / 需要什么字段 / 代码和图片放哪 / 怎么检验模型」一次讲清楚（不需要你改模型）。
+
+---
+
+## 1）模型建立（Model Formulation）需要提供数据吗？
+
+**严格来说：不必。**
+模型建立那一章的任务是：给出变量、假设、方程、机理闭环（功耗映射 + ECM + CPL + 热 + 老化 + 停机判据）。这些可以不靠数据完成。
+
+但有一个例外：
+如果你在模型建立里写了“我们选取 (z_{\min}=0.02)、(\gamma=1.6)、(\kappa=1.2)”这类**具体数值**，那评委会自然追问“这些数怎么来的”。
+所以建议做法是：
+
+* **模型建立**：只写“参数待识别/可由实验/日志获得”，不在这一节塞一堆数值；
+* **参数辨识（第6节）**：再用数据主题/字段说明这些参数怎样拟合；
+* **结果（第8节）**：才给“最终一套参数表 + 范围”。
+
+---
+
+## 2）哪些模块“需要数据”？
+
+你们现在的整篇，真正需要“数据支撑”的地方主要有四块（按重要性）：
+
+### A. 参数辨识（第6节）——最需要数据
+
+你要证明：参数不是拍脑袋，是可以从可复现数据得到。
+
+### B. 结果与验证（第8节/第9节）——需要“对比数据”或“合理性检验”
+
+不一定要真实手机数据，但至少要有：
+
+* 典型输入曲线（场景化）；
+* 对比：仿真输出 vs 常识/基准（比如同类手机续航范围、温升范围）。
+
+### C. 不确定性/MC/Sobol（第7节）——需要“分布假设”的依据
+
+如果你说“亮度是 OU 过程”，那 OU 的均值/方差/相关时间最好来自：
+
+* 历史使用日志统计（最好），或
+* 你给出合理范围并做敏感性（也能过）。
+
+### D. 策略建议（降频/限流）——需要阈值/触发条件（可用假设范围）
+
+比如 (I_{\max}(T_b)) 的阈值、降频比例，可用行业常识范围/简化设定，再做敏感性说明其影响。
+
+---
+
+## 3）如果要提供数据：分别需要哪些“数据主题”？每个主题要哪些字段？
+
+下面给你一套“最小可用数据清单”（MCM 很吃这一套：字段明确、可复现、不会让你真去做重实验也能写得像真的）。
+
+> 你不一定全都有。没有真实数据时，用**合成数据 + 合理范围 + 可复现生成规则**也可以，但字段设计要完整。
+
+---
+
+### 主题 1：系统使用日志（最关键，驱动输入 (\mathbf{u}(t))）
+
+**用途：** 给 (L(t),C(t),N(t),\Psi(t),T_a(t)) 的统计/场景；也可给功耗映射拟合。
+
+**字段建议：**
+
+* `timestamp`（秒或毫秒）
+* `L`（归一化亮度 0–1）
+* `C`（CPU 负载 0–1，或 CPU utilization）
+* `N`（网络活动 0–1：可用吞吐/包率归一化）
+* `Psi`（信号质量：RSRP/RSRQ/SINR 或 0–1 归一化指标）
+* `Ta`（环境温度，℃或 K）
+* 可选：`screen_on`（0/1，用于工况识别）、`app_category`（视频/游戏/社交）
+
+---
+
+### 主题 2：电池测量数据（输出验证 + ECM 拟合）
+
+**用途：** 验证 (V_{\text{term}}(t))、SOC、温度；或拟合 (R_0,R_1,C_1)。
+
+**字段建议：**
+
+* `timestamp`
+* `V_term`（端电压）
+* `I_meas`（电流，若能拿到最好；拿不到也可用功耗+电压反推）
+* `z_meas`（SOC/电量百分比）
+* `Tb_meas`（电池温度）
+* `device_state`（充/放电、是否插电）
+
+---
+
+### 主题 3：OCV–SOC 曲线（拟合 (E_0,K,A,B)）
+
+**用途：** 识别 modified Shepherd 参数。
+
+**字段建议：**
+
+* `z`（0–1）
+* `V_oc`（开路电压）
+* 可选：`temperature`（因为 OCV 也会随温度略变）
+
+---
+
+### 主题 4：脉冲测试数据（拟合 (R_0,R_1,C_1)）
+
+**用途：** 用“瞬时压降 + 指数松弛”分离欧姆/极化。
+
+**字段建议：**
+
+* `timestamp`
+* `I_step`（施加电流）
+* `V_term`（高采样电压）
+* `z`（测试 SOC）
+* `Tb`（测试温度）
+
+---
+
+### 主题 5：网络尾耗数据（拟合 (k_{\text{tail}},\tau_\uparrow,\tau_\downarrow)）
+
+**用途：** 验证你们的 (w(t)) 连续尾耗机制。
+
+**字段建议：**
+
+* `timestamp`
+* `N`（网络活动）
+* `P_net` 或 `P_tot`（网络功耗或整机功耗）
+* `Psi`（信号质量，便于把惩罚项分离）
+* `w`（不需要测，仿真内部；但实验功耗曲线用于拟合）
+
+---
+
+### 主题 6：老化数据（拟合 (\lambda_{\rm sei},m,E_{\rm sei})）
+
+**用途：** 给 SOH 退化方程参数。
+
+**字段建议：**
+
+* `time_or_cycles`
+* `S`（SOH/容量保持率）
+* `I_profile`（平均电流或 C-rate）
+* `Tb`（温度）
+* 可选：`calendar_time`（日历老化）
+
+---
+
+### 主题 7：策略阈值数据（限流/降频）
+
+**用途：** 给 (I_{\max}(T_b)) 或 (P_{\cap}(T_b)) 的形状/阈值。
+
+**字段建议：**
+
+* `Tb`
+* `I_limit` 或 `P_cap`
+* `throttle_flag`（是否触发降频）
+
+> 没有也没关系：用“典型阈值范围 + 敏感性分析”就能写得很合理。
+
+---
+
+## 4）代码应该出现在论文哪个模块？
+
+MCM 论文一般不贴大段代码。推荐结构：
+
+* **第6节 数值求解与参数辨识：** 放“算法伪代码”（Algorithm 3/4）+ 关键实现要点（RK4 子步嵌套求 (I)、步长对半、事件检测）。
+* **附录 Appendix：** 放简化代码框架（比如 Python/Matlab 伪代码），例如：
+
+  * `simulate_one_path(u(t), params)`
+  * `solve_current_CPL()`
+  * `step_RK4_with_nested_I()`
+  * `estimate_params_pulse()`
+  * `MC_TTE()`
+  * `Sobol_Saltelli()`
+
+正文里只要“算法”就够，附录放代码能加分且不抢篇幅。
+
+---
+
+## 5）图片应该出现在哪个模块？
+
+按你们的叙事链条放：
+
+* **第5节 模型建立**：
+
+  * 模型结构框图（输入→功耗→CPL→电流→ODE→输出）
+  * 子系统示意（网络尾耗 (w) 机制图也行）
+
+* **第8节 结果与讨论**（最核心）：
+
+  * 输入样例（(L,C,N,\Psi,T_a)）
+  * SOC/电压/温度轨迹束（MC）
+  * TTE 分布（直方图/CDF/生存曲线）
+  * Sobol 指数条形图
+    -（可选）策略前后对比图（有限流 vs 无）
+
+* **附录**：
+
+  * 参数拟合曲线（OCV 拟合、脉冲松弛拟合、尾耗指数拟合）
+  * 误差收敛图（步长对半、MC 采样收敛）
+
+---
+
+## 6）你的模型如何检验（Validation / Verification）？
+
+这部分你要写得“像工程论文”，我给你一套评委很买账的四层检验法：**V&V（verification & validation）**。
+
+### 6.1 数值正确性（Verification）
+
+证明“代码没算错”：
+
+1. **步长对半收敛**：你们已有
+   (|z_{\Delta t}-z_{\Delta t/2}|_\infty<10^{-4})，TTE 变化 <1%
+2. **物理约束检验**：
+
+   * (z(t)) 单调下降（放电时）
+   * (S(t)) 单调不增
+   * (w(t)\in[0,1])
+3. **能量一致性检查（推荐加分项）**：定义交付能量
+   [
+   E_{\rm del}(t)=\int_0^t V_{\rm term}(\tau)I(\tau),d\tau
+   ]
+   与 SOC 下降对应的可用能量规模一致（不要求完全相等，但量级要对得上）。
+
+### 6.2 参数合理性（Calibration sanity）
+
+证明“参数不离谱”：
+
+* (R_0) 随温度降低增大（Arrhenius）
+* (\gamma,\eta>1) 体现超线性负载
+* (\kappa>0) 表明信号差更耗电
+* (hA/C_{\rm th}) 给出合理热时间常数（几十分钟量级通常合理）
+
+### 6.3 行为有效性（Behavioral validation）
+
+不一定要真实数据，也能做：
+
+* 场景测试：视频/游戏/待机输入，得到 TTE 排序符合常识（游戏 < 视频 < 浏览 < 待机）
+* 低温测试：同样输入下 TTE 明显下降（因为 (R_0\uparrow, Q_{\rm eff}\downarrow)）
+* 老化测试：(S) 下降时 TTE 下降、且更易触发低电压关机
+
+### 6.4 对比验证（External validation）
+
+如果有一点点真实数据最好（哪怕公开数据/文献图）：
+
+* 用文献中“某负载功耗范围、温升范围、续航区间”做对比，证明你输出落在合理区间
+* 或者用手机系统导出的 battery log 做一段短时对比（10–30 分钟也足够）
+
+---
+
+## 你现在应该怎么做（最省力的落地路线）
+
+如果你不想真的去找很多实验数据，我建议“最小闭环”是：
+
+1. **合成/场景化输入数据**（主题1）：自己生成 (L,C,N,\Psi,T_a) 的典型曲线（每个场景 2–4 小时）。
+2. **给一套参数表 + 合理范围**：参数辨识写“可按以下数据拟合”，再给出“采用文献/经验的标称值+范围”，并在敏感性里证明结论稳健。
+3. **结果章用 MC + Sobol + 策略对比**，会非常像 O 奖文章。
+
+---
+
+如果你愿意，我可以把你这段“数据需求说明”直接写成论文里一个小节（例如 **“Data Requirements and Validation Protocol”**），并且帮你把每个数据主题写成表格（字段、单位、来源、用途、对应参数）。你只要告诉我：你们打算走 **真实日志** 还是 **场景合成** 路线即可。
+
+