From dbb62485a760762d55e806dcc7b912cf033e9f64 Mon Sep 17 00:00:00 2001 From: ChuXun <70203584+ChuXunYu@users.noreply.github.com> Date: Fri, 30 Jan 2026 20:40:15 +0800 Subject: [PATCH] =?UTF-8?q?567=E6=8F=90=E4=BA=A4?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- A题/成文/5模型建立.md | 176 +++++++++++++ A题/成文/6数值求解与参数辨识.md | 392 +++++++++++++++++++++++++++ A题/成文/7不确定性建模与统计推断.md | 393 ++++++++++++++++++++++++++++ 3 files changed, 961 insertions(+) create mode 100644 A题/成文/5模型建立.md create mode 100644 A题/成文/6数值求解与参数辨识.md create mode 100644 A题/成文/7不确定性建模与统计推断.md diff --git a/A题/成文/5模型建立.md b/A题/成文/5模型建立.md new file mode 100644 index 0000000..0f80887 --- /dev/null +++ b/A题/成文/5模型建立.md @@ -0,0 +1,176 @@ +\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. diff --git a/A题/成文/6数值求解与参数辨识.md b/A题/成文/6数值求解与参数辨识.md new file mode 100644 index 0000000..0a25a18 --- /dev/null +++ b/A题/成文/6数值求解与参数辨识.md @@ -0,0 +1,392 @@ +% ========================= +% 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)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} diff --git a/A题/成文/7不确定性建模与统计推断.md b/A题/成文/7不确定性建模与统计推断.md new file mode 100644 index 0000000..84dcc2a --- /dev/null +++ b/A题/成文/7不确定性建模与统计推断.md @@ -0,0 +1,393 @@ +%=========================================================== +\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}