## 2 Models and Methods
### DP Algorithm with Emission Constraints
\(\Delta n_{\rm e}\) is the opposite of the difference between the engine speed corresponding to the lth action at the \(j\)-\(k\) grid point and the engine speed corresponding to the \(SOC_{\rm next}\) state in the \((k+1)_{\rm th}\) time step, and the latter can be obtained by interpolating the action information stored in the \((k+1)_{\rm th}\) time step. At the \(j\)-\(k\) grid point, the pollutant emission value loss functions associated with the \(l\)th action are denoted as  
\[J_{\rm CO}(j,k) = [m_{\rm CO}(j,k,l)+J_{\rm CO\_next}] \tag{15}\] \[J_{\rm PN}(j,k) = [m_{\rm PN}(j,k,l)+J_{\rm PN\_next}]\] (16) \[J_{\rm NOx}(j,k) = [m_{\rm NOx}(j,k,l)+J_{\rm NOx\_next}] \tag{17}\]  
where \(J_{\rm CO}(j,k)\), \(J_{\rm PN}(j,k)\), and \(J_{\rm NOx}(j,k)\) are the value loss functions for CO, PN, and \(\rm NOx\) emissions at the \(j\)-\(k\) grid point, respectively. In essence, \(J_{\rm CO}(j,k)\), \(J_{\rm PN}(j,k)\), and \(J_{\rm NOx}(j,k)\) are the cumulative pollutant emissions when the minimum cumulative fuel consumption in the \(k\) to \(N\) time control stage is achieved, with j as the initial SOC state. In these equations, \(m_{\rm CO}(j,k,l)\), \(m_{\rm PN}(j,k,l)\), and \(m_{\rm NOx}(j,k,l)\) are CO, PN, and \(\rm NOx\) emissions due to the \(l\)th action at the \(j\)-\(k\) grid point, calculated by the interpolated-startup-corrected emission model; \(J_{\rm CO\_next}\), \(J_{\rm FN\_next}\), and \(J_{\rm NOx\_next}\) are the CO, PN, and \(\rm NOx\) emission value loss functions under the \(SOC_{\rm next}\) state in the \((k+1)_{\rm th}\) time step, obtained by interpolating the emission value loss functions at the two grid points neighboring \(SOC_{\rm next}\) in the \((k+1)_{\rm th}\) time step. For the \(l\)th action at the \(j\)-\(k\) grid point, the positive relative spatial distance between the cumulative pollutant emissions in the \(k\) to \(N\) time control stage and the pollutant emission limits are  
\[e_{\rm lim}(j,k,l)=\left\{[e_{\rm limCO}(j,k,l)]^{2}+[e_{\rm limPN}(j,k,l)]^{2 }+[e_{\rm limNOx}(j,k,l)]^{2}\right\}^{\frac{1}{2}} \tag{18}\]  
where \(e_{\rm lim}(j,k,l)\) is the positive relative spatial distance for the \(l\)th action at the \(j\)-\(k\) grid point; \(e_{\rm limCO}(j,k,l)\), \(e_{\rm limPN}(j,k,l)\), and \(e_{\rm limNOx}(j,k,l)\) are the components of \(e_{\rm lim}(j,k,l)\) on the CO, PN, and \(\rm NOx\) emission components, respectively, which are  
\[e_{\rm limCO}(j,k,l)=\left\{ \begin{array}{ccc}\frac{J_{\rm CO}(j,k)-J_{\rm CO }{\rm CO}{\rm D}_{\rm trip}(k,N)}{\mu_{\rm CO}{\rm CO}_{\rm D}{\rm trip}(k,N)}&if &J_{\rm CO}(j,k)>\mu_{\rm CO}{\rm e}_{\rm CO}D_{\rm trip}(k,N)\\ 0&if&J_{\rm CO}(j,k)\leq\mu_{\rm CO}{\rm e}_{\rm CO}D_{\rm trip}(k,N)\end{array} \right. \tag{19}\]  
\[e_{\rm limPN}(j,k,l)=\left\{ \begin{array}{ccc}\frac{l_{\rm PN}(j,k)-\mu_{ \rm PN}{\rm PN}{\rm PN}{\rm D}_{\rm trip}(k,N)}{\mu_{\rm FN}{\rm PN}{\rm D}_{ \rm trip}(k,N)}&if&J_{\rm PN}(j,k)>\mu_{\rm FN}{\rm per}{\rm PN}{\rm D}_{\rm trip }(k,N)\\ 0&if&J_{\rm PN}(j,k)\leq\mu_{\rm FN}{\rm per}{\rm PN}{\rm D}_{\rm trip}(k,N) \end{array} \right. \tag{20}\]  
\[e_{\rm limNOx}(j,k,l)=\left\{ \begin{array}{ccc}\frac{l_{\rm NOx}(j,k)-\mu_{ \rm NOx}{\rm NOx}_{\rm D}{\rm trip}(k,N)}{\mu_{\rm NOx}{\rm NOx}_{\rm D}{ \rm trip}(k,N)}&if&J_{\rm NOx}(j,k)>\mu_{\rm NOx}{\rm e}_{\rm NOx}D_{\rm trip }(k,N)\\ 0&if&J_{\rm NOx}(j,k)\leq\mu_{\rm NOx}{\rm e}_{\rm NOx}D_{\rm trip}(k,N)\end{array} \right. \tag{21}\]  
where \(D_{\rm trip}(k,N)\) is the distance traveled by the vehicle in the \(k\) to \(N\) period; \(\mu_{\rm CO}\), \(\mu_{\rm PN}\), and \(\mu_{\rm NOx}\) are the relaxation factors for the pollutant emission limits at the intermediate time steps, which are co-adjusted with \(\rm m_{\rm o}\) to make the emission limits for the different pollutant emission factors satisfied in the total trip. Up to this point, \(m_{\rm fuel}(j,k,l)\), \(I_{\rm fuel,next}\), and \(e_{\rm lim}(j,k,l)\) in Equation (11) are calculated.