Model Mathematics

tau = \{\begin{matrix} time if time ≦ period \\ time - period if time ≦ period 2 \\ time - (period 2) if time ≦ period 3 \\ time - (period 3) if time ≦ period 4 \\ time - (period 4) if time ≦ period 5 \\ time - (period 5) if time ≦ period 6 \\ time - (period 6) if time ≦ period 7 \\ time - (period 7) if time ≦ period 8 \\ time - (period 8) if time ≦ period 9 \\ time - (period 9) if time ≦ period 10 \\ time - (period 10) if time ≦ period 11 \\ time - (period 11) if time ≦ period 12 \\ time - (period 12) if time ≦ period 13 \end{matrix} e_t = A ⅇ^{(- B) {(tau - C)}^{2}}

V_pcd = V_lv + V_rv P_pcd = P_0_pcd (ⅇ^{lambda_pcd (V_pcd - V_0_pcd)} - 1) P_peri = P_pcd + P_th

V_lvf = V_lv - V_spt P_lvf = e_t P_es_lvf + (1 - e_t) P_ed_lvf P_lv = P_lvf + P_peri \frac{d}{d time} (V_lv) = \{\begin{matrix} 0 if Q_mt < 0 \land Q_av < 0 \\ - Q_av if Q_mt < 0 \\ Q_mt if Q_av < 0 \\ Q_mt - Q_av otherwise \end{matrix}

P_es_lvf = E_es_lvf (V_lvf - V_d_lvf) P_ed_lvf = P_0_lvf (ⅇ^{lambda_lvf (V_lvf - V_0_lvf)} - 1)

V_rvf = V_rv + V_spt P_rvf = e_t P_es_rvf + (1 - e_t) P_ed_rvf P_rv = P_rvf + P_peri \frac{d}{d time} (V_rv) = \{\begin{matrix} 0 if Q_tc < 0 \land Q_pv < 0 \\ - Q_pv if Q_tc < 0 \\ Q_tc if Q_pv < 0 \\ Q_tc - Q_pv otherwise \end{matrix}

P_es_rvf = E_es_rvf (V_rvf - V_d_rvf) P_ed_rvf = P_0_rvf (ⅇ^{lambda_rvf (V_rvf - V_0_rvf)} - 1)

0 = (e_t E_es_spt (V_spt - V_d_spt) + (one - e_t) P_0_spt (ⅇ^{lambda_spt (V_spt - V_0_spt)} - one) - (e_t E_es_lvf (V_lv - V_spt)) - ((1 - e_t) P_0_lvf (ⅇ^{lambda_lvf (V_lv - V_spt)} - 1))) + e_t E_es_rvf (V_rv + V_spt) + (1 - e_t) P_0_rvf (ⅇ^{lambda_rvf (V_rv + V_spt)} - 1)

P_pa = E_es_pa (V_pa - V_d_pa) + P_th \frac{d}{d time} (V_pa) = \{\begin{matrix} - Q_pul if Q_pv < 0 \\ Q_pv - Q_pul otherwise \end{matrix}

P_pu = E_es_pu (V_pu - V_d_pu) + P_th \frac{d}{d time} (V_pu) = \{\begin{matrix} Q_pul if Q_mt < 0 \\ Q_pul - Q_mt otherwise \end{matrix}

P_ao = E_es_ao (V_ao - V_d_ao) \frac{d}{d time} (V_ao) = \{\begin{matrix} - Q_sys if Q_av < 0 \\ Q_av - Q_sys otherwise \end{matrix}

P_vc = E_es_vc (V_vc - V_d_vc) \frac{d}{d time} (V_vc) = \{\begin{matrix} Q_sys if Q_tc < 0 \\ Q_sys - Q_tc otherwise \end{matrix}

Q_sys = \frac{P_ao - P_vc}{R_sys} Q_pul = \frac{P_pa - P_pu}{R_pul} \frac{d}{d time} (Q_mt) = \{\begin{matrix} 0 if P_pu - P_lv < 0 \land Q_mt < 0 \\ \frac{P_pu - P_lv - (Q_mt R_mt)}{L_mt} otherwise \end{matrix} \frac{d}{d time} (Q_av) = \{\begin{matrix} 0 if P_lv - P_ao < 0 \land Q_av < 0 \\ \frac{P_lv - P_ao - (Q_av R_av)}{L_av} otherwise \end{matrix} \frac{d}{d time} (Q_tc) = \{\begin{matrix} 0 if P_vc - P_rv < 0 \land Q_tc < 0 \\ \frac{P_vc - P_rv - (Q_tc R_tc)}{L_tc} otherwise \end{matrix} \frac{d}{d time} (Q_pv) = \{\begin{matrix} 0 if P_rv - P_pa < 0 \land Q_pv < 0 \\ \frac{P_rv - P_pa - (Q_pv R_pv)}{L_pv} otherwise \end{matrix}