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fix for numerical runaway of rounding errors

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due to turbine-pipeline interatction
via a convergence method in the turbine
and a "damping" trick on the reservoir velocity
plus: code cleanup with consistent naming of variables
This commit is contained in:
Brantegger Georg
2022-08-03 15:56:56 +02:00
parent 84631ee4cc
commit ba696444bb
13 changed files with 1257 additions and 1198 deletions
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176
Druckrohrleitung/Druckrohrleitung_test_steady_state.ipynb
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@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
"execution_count": 9,
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
@@ -22,102 +22,103 @@
},
{
"cell_type": "code",
"execution_count": 10,
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib qt5\n",
"#define constants pipe\n",
"# define constants\n",
"\n",
"g = 9.81 # gravitational acceleration [m/s²]\n",
"rho = 1000. # density of water [kg/m³]\n",
"\n",
"L = 1000. # length of pipeline [m]\n",
"D = 0.9 # pipe diameter [m]\n",
"h_res = 10. # water level in upstream reservoir [m]\n",
"n = 50 # number of pipe segments in discretization\n",
"nt = 5000 # number of time steps after initial conditions\n",
"f_D = 0.01 # Darcy friction factor\n",
"c = 400. # propagation velocity of the pressure wave [m/s]\n",
"h_pipe = 105. # hydraulic head without reservoir [m] \n",
"alpha = np.arcsin(h_pipe/L) # Höhenwinkel der Druckrohrleitung \n",
" # for physics\n",
"g = 9.81 # [m/s²] gravitational acceleration \n",
"rho = 1000. # [kg/m³] density of water \n",
"pUnit_calc = 'Pa' # [text] DO NOT CHANGE! for pressure conversion in print statements and plot labels \n",
"pUnit_conv = 'mWS' # [text] for pressure conversion in print statements and plot labels\n",
"\n",
"\n",
"# preparing the discretization and initial conditions\n",
"initial_flux = 0.8 # m³/s\n",
"initial_level = h_res # m\n",
"dx = L/n # length of each pipe segment\n",
"dt = dx/c # timestep according to method of characterisitics\n",
"nn = n+1 # number of nodes\n",
"pl_vec = np.arange(0,nn,1)*dx # pl = pipe-length. position of the nodes on the pipeline\n",
"t_vec = np.arange(0,nt,1)*dt # time vector\n",
"h_vec = np.arange(0,nn,1)*h_pipe/n # hydraulic head of pipeline at each node\n",
" # for Turbine\n",
"Tur_Q_nenn = 0.85 # [m³/s] nominal flux of turbine \n",
"Tur_p_nenn = pressure_conversion(10.6,'bar',pUnit_calc) # [Pa] nominal pressure of turbine \n",
"Tur_closingTime = 90. # [s] closing time of turbine\n",
"\n",
"\n",
"# define constants reservoir\n",
"conversion_pressure_unit = 'mWS'\n",
" # for PI controller\n",
"Con_targetLevel = 8. # [m]\n",
"Con_K_p = 0.1 # [-] proportional constant of PI controller\n",
"Con_T_i = 10. # [s] timespan in which a steady state error is corrected by the intergal term\n",
"Con_deadbandRange = 0.05 # [m] Deadband range around targetLevel for which the controller does NOT intervene\n",
"\n",
"area_base = 75. # m²\n",
"area_pipe = (D/2)**2*np.pi # m²\n",
"critical_level_low = 0. # m\n",
"critical_level_high = 100. # m\n",
"\n",
"# make sure e-RK4 method of reservoir has a small enough timestep to avoid runaway numerical error\n",
"nt_eRK4 = 1 # number of simulation steps of reservoir in between timesteps of pipeline \n",
"simulation_timestep = dt/nt_eRK4"
" # for pipeline\n",
"Pip_length = (535.+478.) # [m] length of pipeline\n",
"Pip_dia = 0.9 # [m] diameter of pipeline\n",
"Pip_area = Pip_dia**2/4*np.pi # [m²] crossectional area of pipeline\n",
"Pip_head = 105. # [m] hydraulic head of pipeline without reservoir\n",
"Pip_angle = np.arcsin(Pip_head/Pip_length) # [rad] elevation angle of pipeline \n",
"Pip_n_seg = 50 # [-] number of pipe segments in discretization\n",
"Pip_f_D = 0.014 # [-] Darcy friction factor\n",
"Pip_pw_vel = 500. # [m/s] propagation velocity of the pressure wave (pw) in the given pipeline\n",
" # derivatives of the pipeline constants\n",
"Pip_dx = Pip_length/Pip_n_seg # [m] length of each pipe segment\n",
"Pip_dt = Pip_dx/Pip_pw_vel # [s] timestep according to method of characteristics\n",
"Pip_nn = Pip_n_seg+1 # [1] number of nodes\n",
"Pip_x_vec = np.arange(0,Pip_nn,1)*Pip_dx # [m] vector holding the distance of each node from the upstream reservoir along the pipeline\n",
"Pip_h_vec = np.arange(0,Pip_nn,1)*Pip_head/Pip_n_seg # [m] vector holding the vertival distance of each node from the upstream reservoir\n",
"\n",
"\n",
" # for reservoir\n",
"Res_area_base = 74. # [m²] total base are of the cuboid reservoir \n",
"Res_area_out = Pip_area # [m²] outflux area of the reservoir, given by pipeline area\n",
"Res_level_crit_lo = 0. # [m] for yet-to-be-implemented warnings\n",
"Res_level_crit_hi = np.inf # [m] for yet-to-be-implemented warnings\n",
"Res_dt_approx = 1e-3 # [s] approx. timestep of reservoir time evolution to ensure numerical stability (see Res_nt why approx.)\n",
"Res_nt = max(1,int(Pip_dt//Res_dt_approx)) # [1] number of timesteps of the reservoir time evolution within one timestep of the pipeline\n",
"Res_dt = Pip_dt/Res_nt # [s] harmonised timestep of reservoir time evolution\n",
"\n",
" # for general simulation\n",
"flux_init = Tur_Q_nenn/1.1 # [m³/s] initial flux through whole system for steady state initialization \n",
"level_init = Con_targetLevel # [m] initial water level in upstream reservoir for steady state initialization\n",
"simTime_target = 600. # [s] target for total simulation time (will vary slightly to fit with Pip_dt)\n",
"nt = int(simTime_target//Pip_dt) # [1] Number of timesteps of the whole system\n",
"t_vec = np.arange(0,nt+1,1)*Pip_dt # [s] time vector. At each step of t_vec the system parameters are stored\n"
]
},
{
"cell_type": "code",
"execution_count": 11,
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"V = Ausgleichsbecken_class(area_base,area_pipe,critical_level_low,critical_level_high,simulation_timestep)\n",
"V.set_steady_state(initial_flux,initial_level,conversion_pressure_unit)\n",
"# create objects\n",
"\n",
"pipe = Druckrohrleitung_class(L,D,n,alpha,f_D)\n",
"pipe.set_pressure_propagation_velocity(c)\n",
"pipe.set_number_of_timesteps(nt)\n",
"pipe.set_steady_state(initial_flux,initial_level,area_base,pl_vec,h_vec)"
"# Upstream reservoir\n",
"reservoir = Ausgleichsbecken_class(Res_area_base,Res_area_out,Res_dt,Res_level_crit_lo,Res_level_crit_hi,rho)\n",
"reservoir.set_steady_state(flux_init,level_init)\n",
"\n",
"# pipeline\n",
"pipe = Druckrohrleitung_class(Pip_length,Pip_dia,Pip_n_seg,Pip_angle,Pip_f_D,Pip_pw_vel,Pip_dt,pUnit_conv,rho)\n",
"pipe.set_steady_state(flux_init,level_init,Res_area_base,Pip_x_vec,Pip_h_vec)\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"execution_count": null,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The current attributes are: \n",
"----------------------------- \n",
"Current level = 10.0 m\n",
"Volume in reservoir = -- m³ \n",
"Current influx = 0.8 m³/s \n",
"Current outflux = 0.8 m³/s \n",
"Current outflux vel = 1.258 m/s \n",
"Current pipe pressure = 9.844 mWS \n",
"----------------------------- \n",
"\n"
]
}
],
"outputs": [],
"source": [
"V.get_info()"
"reservoir.get_info(full=True)\n",
"pipe.get_info(full=True)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"# initialization for timeloop\n",
"\n",
"level_vec = np.zeros_like(t_vec)\n",
"level_vec[0] = V.get_current_level()\n",
"level_vec[0] = reservoir.get_current_level()\n",
"\n",
"# prepare the vectors in which the pressure and velocity distribution in the pipeline from the previous timestep are stored\n",
"v_old = pipe.get_current_velocity_distribution()\n",
@@ -141,21 +142,23 @@
},
{
"cell_type": "code",
"execution_count": 14,
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib qt5\n",
"fig1,axs1 = plt.subplots(2,1)\n",
"axs1[0].set_title('Pressure distribution in pipeline')\n",
"axs1[0].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs1[0].set_ylabel(r'$p$ [mWS]')\n",
"lo_00, = axs1[0].plot(pl_vec,pressure_conversion(p_old,'Pa',conversion_pressure_unit),marker='.')\n",
"axs1[0].set_ylim([0.9*np.min(pressure_conversion(p_old,'Pa',conversion_pressure_unit)),1.1*np.max(pressure_conversion(p_old,'Pa',conversion_pressure_unit))])\n",
"axs1[0].set_ylim([0.9*np.min(pressure_conversion(p_old,'Pa',pUnit_conv)),1.1*np.max(pressure_conversion(p_old,'Pa',pUnit_conv))])\n",
"lo_00, = axs1[0].plot(Pip_x_vec,pressure_conversion(p_old,'Pa',pUnit_conv),marker='.')\n",
"\n",
"axs1[1].set_title('Velocity distribution in pipeline')\n",
"axs1[1].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs1[1].set_ylabel(r'$v$ [m/s]')\n",
"lo_01, = axs1[1].plot(pl_vec,v_old,marker='.')\n",
"lo_01, = axs1[1].plot(Pip_x_vec,v_old,marker='.')\n",
"axs1[1].autoscale()\n",
"# axs1[1].set_ylim([0.9*np.min(v_old),1.1*np.max(v_boundary_res)])\n",
"\n",
"fig1.tight_layout()\n",
@@ -164,25 +167,24 @@
},
{
"cell_type": "code",
"execution_count": 15,
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"\n",
"for it_pipe in range(1,nt):\n",
"for it_pipe in range(1,nt+1):\n",
"# for each pipeline timestep, execute nt_eRK4 timesteps of the reservoir code\n",
" # set initial conditions for the reservoir time evolution calculted with e-RK4\n",
" V.set_pressure(p_old[0])\n",
" V.set_outflux(v_old[0]*area_pipe)\n",
" reservoir.set_pressure(p_old[0],display_warning=False)\n",
" reservoir.set_outflux(v_old[0]*Pip_area,display_warning=False)\n",
" # calculate the time evolution of the reservoir level within each pipeline timestep to avoid runaway numerical error\n",
" for it_res in range(nt_eRK4):\n",
" V.timestep_reservoir_evolution() \n",
" level_vec[it_pipe] = V.get_current_level() \n",
" for it_res in range(Res_nt):\n",
" reservoir.timestep_reservoir_evolution() \n",
" level_vec[it_pipe] = reservoir.get_current_level() \n",
"\n",
" \n",
" # set boundary conditions for the next timestep of the characteristic method\n",
" p_boundary_res[it_pipe] = V.get_current_pressure()\n",
" v_boundary_tur[it_pipe] = initial_flux/area_pipe\n",
" p_boundary_res[it_pipe] = reservoir.get_current_pressure()\n",
" v_boundary_tur[it_pipe] = flux_init/Pip_area\n",
"\n",
" # the the boundary conditions in the pipe.object and thereby calculate boundary pressure at turbine\n",
" pipe.set_boundary_conditions_next_timestep(p_boundary_res[it_pipe],v_boundary_tur[it_pipe])\n",
@@ -202,28 +204,30 @@
" lo_01.remove()\n",
" # lo_02.remove()\n",
" # plot new pressure and velocity distribution in the pipeline\n",
" lo_00, = axs1[0].plot(pl_vec,pressure_conversion(p_old,'Pa', conversion_pressure_unit),marker='.',c='blue')\n",
" lo_01, = axs1[1].plot(pl_vec,v_old,marker='.',c='blue')\n",
" lo_00, = axs1[0].plot(Pip_x_vec,pressure_conversion(p_old,'Pa', pUnit_conv),marker='.',c='blue')\n",
" lo_01, = axs1[1].plot(Pip_x_vec,v_old,marker='.',c='blue')\n",
" \n",
" fig1.suptitle(str(round(t_vec[it_pipe],2)) + '/' + str(round(t_vec[-1],2)))\n",
" fig1.canvas.draw()\n",
" fig1.tight_layout()\n",
" plt.pause(0.000001)\n",
"\n"
"\n",
"reservoir.get_info(full=True)\n",
"pipe.get_info(full=True)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"fig2,axs2 = plt.subplots(2,2)\n",
"axs2[0,0].set_title('Pressure Reservoir')\n",
"axs2[0,0].plot(t_vec,pressure_conversion(p_boundary_res,'Pa',conversion_pressure_unit))\n",
"axs2[0,0].plot(t_vec,pressure_conversion(p_boundary_res,pUnit_calc,pUnit_conv))\n",
"axs2[0,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[0,0].set_ylabel(r'$p$ [mWS]')\n",
"axs2[0,0].set_ylim([0.9*np.min(pressure_conversion(p_boundary_res,'Pa',conversion_pressure_unit)),1.1*np.max(pressure_conversion(p_boundary_res,'Pa',conversion_pressure_unit))])\n",
"axs2[0,0].set_ylim([0.9*np.min(pressure_conversion(p_boundary_res,pUnit_calc,pUnit_conv)),1.1*np.max(pressure_conversion(p_boundary_res,pUnit_calc,pUnit_conv))])\n",
"\n",
"axs2[0,1].set_title('Velocity Reservoir')\n",
"axs2[0,1].plot(t_vec,v_boundary_res)\n",
@@ -232,16 +236,16 @@
"axs2[0,1].set_ylim([0.9*np.min(v_boundary_res),1.1*np.max(v_boundary_res)])\n",
"\n",
"axs2[1,0].set_title('Pressure Turbine')\n",
"axs2[1,0].plot(t_vec,pressure_conversion(p_boundary_tur,'Pa',conversion_pressure_unit))\n",
"axs2[1,0].plot(t_vec,pressure_conversion(p_boundary_tur,pUnit_calc,pUnit_conv))\n",
"axs2[1,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[1,0].set_ylabel(r'$p$ [mWS]')\n",
"axs2[1,0].set_ylim([0.9*np.min(pressure_conversion(p_boundary_tur,'Pa',conversion_pressure_unit)),1.1*np.max(pressure_conversion(p_boundary_tur,'Pa',conversion_pressure_unit))])\n",
"axs2[1,0].set_ylim([0.9*np.min(pressure_conversion(p_boundary_tur,pUnit_calc,pUnit_conv)),1.1*np.max(pressure_conversion(p_boundary_tur,pUnit_calc,pUnit_conv))])\n",
"\n",
"axs2[1,1].set_title('Velocity Turbine')\n",
"axs2[1,1].plot(t_vec,v_boundary_tur)\n",
"axs2[1,1].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[1,1].set_ylabel(r'$v$ [$\\mathrm{m}/\\mathrm{s}$]')\n",
"axs2[1,1].set_ylim([0.9*np.min(v_boundary_tur),1.1*np.max(v_boundary_tur)])\n",
"axs2[1,1].set_ylim([0.95*np.min(v_boundary_tur),1.05*np.max(v_boundary_tur)])\n",
"\n",
"fig2.tight_layout()\n",
"plt.show()"