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Merge branch 'Dev'

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Brantegger Georg
2022-08-08 07:53:06 +02:00
parent 1aee8debc4 5a790d5ca5
commit 112c9d427a
11 changed files with 1432 additions and 856 deletions
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131
Turbinen/Turbinen_class_file.py
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@@ -1,8 +1,10 @@
from time import time
import numpy as np
#importing pressure conversion function
import sys
import os
from pyparsing import alphanums
current = os.path.dirname(os.path.realpath(__file__))
parent = os.path.dirname(current)
sys.path.append(parent)
@@ -11,7 +13,7 @@ from functions.pressure_conversion import pressure_conversion
class Francis_Turbine:
# units
# make sure that units and print units are the same
# units are used to label graphs and print units are used to have a bearable format when using pythons print()
# units are used to label graphs and disp units are used to have a bearable format when using pythons print()
density_unit = r'$\mathrm{kg}/\mathrm{m}^3$'
flux_unit = r'$\mathrm{m}^3/\mathrm{s}$'
LA_unit = '%'
@@ -20,30 +22,28 @@ class Francis_Turbine:
velocity_unit = r'$\mathrm{m}/\mathrm{s}$'
volume_unit = r'$\mathrm{m}^3$'
density_unit_print = 'kg/m³'
flux_unit_print = 'm³/s'
LA_unit_print = '%'
pressure_unit_print = 'mWS'
time_unit_print = 's'
velocity_unit_print = 'm/s'
volume_unit_print = 'm³'
density_unit_disp = 'kg/m³'
flux_unit_disp = 'm³/s'
LA_unit_disp = '%'
time_unit_disp = 's'
velocity_unit_disp = 'm/s'
volume_unit_disp = 'm³'
g = 9.81 # m/s² gravitational acceleration
# init
def __init__(self, Q_nenn,p_nenn,t_closing=-1.,timestep=-1.):
def __init__(self, Q_nenn,p_nenn,t_closing,timestep,pressure_unit_disp):
self.Q_n = Q_nenn # nominal flux
self.p_n = p_nenn # nominal pressure
self.LA_n = 1. # 100% # nominal Leitapparatöffnung
h = pressure_conversion(p_nenn,'Pa','MWs') # nominal pressure in terms of hydraulic head
self.A = Q_nenn/(np.sqrt(2*self.g*h)*0.98) # Ersatzfläche
self.dt = timestep # simulation timestep
self.t_c = t_closing # closing time
self.t_c = t_closing # closing time
self.d_LA_max_dt = 1/t_closing # maximal change of LA per second
self.pressure_unit_disp = pressure_unit_disp
# initialize for get_info() - parameters will be converted to display -1 if not overwritten
self.p = pressure_conversion(-1,self.pressure_unit_print,self.pressure_unit)
self.p = pressure_conversion(-1,self.pressure_unit_disp,self.pressure_unit)
self.Q = -1.
self.LA = -0.01
@@ -54,19 +54,22 @@ class Francis_Turbine:
self.LA = LA
# warn user, that the .set_LA() method should not be used ot set LA manually
if display_warning == True:
print('Consider using the .update_LA() method instead of setting LA manually')
def set_timestep(self,timestep,display_warning=True):
# set Leitapparatöffnung
self.dt = time
# warn user, that the .set_LA() method should not be used ot set LA manually
if display_warning == True:
print('WARNING: You are changing the timestep of the turbine simulation. This has implications on the simulated closing speed!')
print('You are setting the guide vane opening of the turbine manually. \n \
This is not an intended use of this method. \n \
Refer to the .update_LA() method instead.')
def set_pressure(self,pressure):
# set pressure in front of the turbine
self.p = pressure
def set_steady_state(self,ss_flux,ss_pressure):
# calculate and set steady state LA, that allows the flow of ss_flux at ss_pressure through the
# turbine at the steady state LA
ss_LA = self.LA_n*ss_flux/self.Q_n*np.sqrt(self.p_n/ss_pressure)
if ss_LA < 0 or ss_LA > 1:
raise Exception('LA out of range [0;1]')
self.set_LA(ss_LA,display_warning=False)
#getter
def get_current_Q(self):
# return the flux through the turbine, based on the current pressure in front
@@ -80,10 +83,13 @@ class Francis_Turbine:
def get_current_LA(self):
return self.LA
def get_current_pressure(self):
return pressure_conversion(self.p,self.pressure_unit,self.pressure_unit_disp)
def get_info(self, full = False):
new_line = '\n'
p = pressure_conversion(self.p,self.pressure_unit,self.pressure_unit_print)
p_n = pressure_conversion(self.p_n,self.pressure_unit,self.pressure_unit_print)
p = pressure_conversion(self.p,self.pressure_unit,self.pressure_unit_disp)
p_n = pressure_conversion(self.p_n,self.pressure_unit,self.pressure_unit_disp)
if full == True:
@@ -91,33 +97,34 @@ class Francis_Turbine:
print_str = (f"Turbine has the following attributes: {new_line}"
f"----------------------------- {new_line}"
f"Type = Francis {new_line}"
f"Nominal flux = {self.Q_n:<10} {self.flux_unit_print} {new_line}"
f"Nominal pressure = {round(p_n,3):<10} {self.pressure_unit_print}{new_line}"
f"Nominal LA = {self.LA_n*100:<10} {self.LA_unit_print} {new_line}"
f"Closing time = {self.t_c:<10} {self.time_unit_print} {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_print} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_print} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_print} {new_line}"
f"Simulation timestep = {self.dt:<10} {self.time_unit_print} {new_line}"
f"Nominal flux = {self.Q_n:<10} {self.flux_unit_disp} {new_line}"
f"Nominal pressure = {round(p_n,3):<10} {self.pressure_unit_disp}{new_line}"
f"Nominal LA = {self.LA_n*100:<10} {self.LA_unit_disp} {new_line}"
f"Closing time = {self.t_c:<10} {self.time_unit_disp} {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_disp} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_disp} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_disp} {new_line}"
f"Simulation timestep = {self.dt:<10} {self.time_unit_disp} {new_line}"
f"----------------------------- {new_line}")
else:
# :<10 pads the self.value to be 10 characters wide
print_str = (f"The current attributes are: {new_line}"
f"----------------------------- {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_print} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_print} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_print} {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_disp} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_disp} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_disp} {new_line}"
f"----------------------------- {new_line}")
print(print_str)
# methods
# update methods
def update_LA(self,LA_soll):
# update the Leitappartöffnung and consider the restrictions of the closing time of the turbine
LA_diff = self.LA-LA_soll # calculate the difference to the target LA
LA_diff_max = self.d_LA_max_dt*self.dt # calculate the maximum change in LA based on the given timestep
LA_diff = self.LA-LA_soll # calculate the difference to the target LA
LA_diff_max = self.d_LA_max_dt*self.dt # calculate the maximum possible change in LA based on the given timestep
LA_diff = np.sign(LA_diff)*np.min(np.abs([LA_diff,LA_diff_max])) # calulate the correct change in LA
# make sure that the LA is not out of the range [0;1]
LA_new = self.LA-LA_diff
if LA_new < 0.:
LA_new = 0.
@@ -125,10 +132,42 @@ class Francis_Turbine:
LA_new = 1.
self.set_LA(LA_new,display_warning=False)
def set_steady_state(self,ss_flux,ss_pressure):
# calculate and set steady state LA, that allows the flow of ss_flux at ss_pressure through the
# turbine at the steady state LA
ss_LA = self.LA_n*ss_flux/self.Q_n*np.sqrt(self.p_n/ss_pressure)
if ss_LA < 0 or ss_LA > 1:
raise Exception('LA out of range [0;1]')
self.set_LA(ss_LA,display_warning=False)
# methods
def converge(self,convergence_parameters):
# small numerical disturbances (~1e-12 m/s) in the velocity can get amplified at the turbine node, because the new velocity of the turbine and the
# new pressure from the forward characteristic are not compatible.
eps = 1e-12 # convergence criterion: iteration change < eps
iteration_change = 1. # change in Q from one iteration to the next
i = 0 # safety variable. break loop if it exceeds 1e6 iterations
g = self.g # gravitational acceleration
p = convergence_parameters[0] # pressure at second to last node (see method of characterisctics - boundary condidtions)
v = convergence_parameters[1] # velocity at second to last node (see method of characterisctics - boundary condidtions)
D = convergence_parameters[2] # diameter of the pipeline
area_pipe = convergence_parameters[3] # area of the pipeline
alpha = convergence_parameters[4] # elevation angle of the pipeline
f_D = convergence_parameters[5] # Darcy friction coefficient
c = convergence_parameters[6] # pressure wave propagtation velocity
rho = convergence_parameters[7] # density of the liquid
dt = convergence_parameters[8] # timestep of the characteristic method
p_old = self.get_current_pressure()
Q_old = self.get_current_Q()
v_old = Q_old/area_pipe
while iteration_change > eps:
self.set_pressure(p_old)
Q_new = self.get_current_Q()
v_new = Q_new/area_pipe
p_new = p-rho*c*(v_old-v)+rho*c*dt*g*np.sin(alpha)-f_D*rho*c*dt/(2*D)*abs(v)*v
iteration_change = abs(Q_old-Q_new)
Q_old = Q_new.copy()
p_old = p_new.copy()
v_old = v_new.copy()
i = i+1
if i == 1e6:
print('did not converge')
break
# print(i)
self.Q = Q_new

370
Turbinen/Turbinen_test_steady_state.ipynb Normal file
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@@ -0,0 +1,370 @@
{
"cells": [
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from Turbinen_class_file import Francis_Turbine\n",
"\n",
"import sys\n",
"import os\n",
"current = os.path.dirname(os.path.realpath('Main_Programm.ipynb'))\n",
"parent = os.path.dirname(current)\n",
"sys.path.append(parent)\n",
"from functions.pressure_conversion import pressure_conversion\n",
"from Ausgleichsbecken.Ausgleichsbecken_class_file import Ausgleichsbecken_class\n",
"from Druckrohrleitung.Druckrohrleitung_class_file import Druckrohrleitung_class\n",
"from Regler.Regler_class_file import PI_controller_class"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"# define constants\n",
"\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",
" # 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",
" # 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",
"\n",
" # 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": 10,
"metadata": {},
"outputs": [],
"source": [
"# create objects\n",
"\n",
"# 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",
"\n",
"# downstream turbine\n",
"turbine = Francis_Turbine(Tur_Q_nenn,Tur_p_nenn,Tur_closingTime,Pip_dt,pUnit_conv)\n",
"turbine.set_steady_state(flux_init,pipe.get_current_pressure_distribution()[-1])\n",
"\n",
"# influx setting turbine\n",
"turbine_in = Francis_Turbine(Tur_Q_nenn,Tur_p_nenn,Tur_closingTime/2,Pip_dt,pUnit_conv)\n",
"turbine_in.set_steady_state(flux_init,Tur_p_nenn)\n",
"\n",
"# level controll\n",
"level_control = PI_controller_class(Con_targetLevel,Con_deadbandRange,Con_K_p,Con_T_i,Pip_dt)\n",
"level_control.set_control_variable(turbine.get_current_LA(),display_warning=False)\n"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
"# initialization for Timeloop\n",
"\n",
"v_old = pipe.get_current_velocity_distribution()\n",
"v_min = pipe.get_current_velocity_distribution()\n",
"v_max = pipe.get_current_velocity_distribution()\n",
"Q_old = pipe.get_current_flux_distribution()\n",
"Q_min = pipe.get_current_flux_distribution()\n",
"Q_max = pipe.get_current_flux_distribution()\n",
"p_old = pipe.get_current_pressure_distribution()\n",
"p_min = pipe.get_current_pressure_distribution()\n",
"p_max = pipe.get_current_pressure_distribution()\n",
"\n",
"Q_in_vec = np.zeros_like(t_vec)\n",
"Q_in_vec[0] = flux_init\n",
"\n",
"v_boundary_res = np.zeros_like(t_vec)\n",
"v_boundary_tur = np.zeros_like(t_vec)\n",
"Q_boundary_res = np.zeros_like(t_vec)\n",
"Q_boundary_tur = np.zeros_like(t_vec)\n",
"p_boundary_res = np.zeros_like(t_vec)\n",
"p_boundary_tur = np.zeros_like(t_vec)\n",
"\n",
"level_vec = np.full_like(t_vec,level_init) # level at the end of each pipeline timestep\n",
"volume_vec = np.full_like(t_vec,reservoir.get_current_volume()) # volume at the end of each pipeline timestep\n",
"\n",
"v_boundary_res[0] = v_old[0]\n",
"v_boundary_tur[0] = v_old[-1] \n",
"Q_boundary_res[0] = Q_old[0]\n",
"Q_boundary_tur[0] = Q_old[-1]\n",
"p_boundary_res[0] = p_old[0]\n",
"p_boundary_tur[0] = p_old[-1]\n",
"\n",
"LA_soll_vec = np.full_like(t_vec,turbine.get_current_LA())\n",
"LA_ist_vec = np.full_like(t_vec,turbine.get_current_LA())\n",
"\n",
"LA_soll_vec2 = np.full_like(t_vec,turbine_in.get_current_LA())\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib qt5\n",
"# Con_T_ime loop\n",
"\n",
"# create a figure and subplots to display the velocity and pressure distribution across the pipeline in each pipeline step\n",
"fig1,axs1 = plt.subplots(2,1)\n",
"fig1.suptitle(str(0) +' s / '+str(round(t_vec[-1],2)) + ' s' )\n",
"axs1[0].set_title('Pressure distribution in pipeline')\n",
"axs1[0].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs1[0].set_ylabel(r'$p$ ['+pUnit_conv+']')\n",
"axs1[1].set_title('Flux distribution in pipeline')\n",
"axs1[1].set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs1[1].set_ylabel(r'$Q$ [$\\mathrm{m}^3 / \\mathrm{s}$]')\n",
"lo_p, = axs1[0].plot(Pip_x_vec,pressure_conversion(p_old,pUnit_calc, pUnit_conv),marker='.')\n",
"lo_q, = axs1[1].plot(Pip_x_vec,Q_old,marker='.')\n",
"lo_pmin, = axs1[0].plot(Pip_x_vec,pipe.get_lowest_pressure_per_node(disp=True),c='red')\n",
"lo_pmax, = axs1[0].plot(Pip_x_vec,pipe.get_highest_pressure_per_node(disp=True),c='red')\n",
"lo_qmin, = axs1[1].plot(Pip_x_vec,pipe.get_lowest_flux_per_node(),c='red')\n",
"lo_qmax, = axs1[1].plot(Pip_x_vec,pipe.get_highest_flux_per_node(),c='red')\n",
"\n",
"axs1[0].autoscale()\n",
"axs1[1].autoscale()\n",
"\n",
"fig1.tight_layout()\n",
"fig1.show()\n",
"plt.pause(1)\n"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
"convergence_parameters = [p_old[-2],v_old[-2],Pip_dia,Pip_area,Pip_angle,Pip_f_D,Pip_pw_vel,rho,Pip_dt]\n",
"\n",
"# loop through Con_T_ime steps of the pipeline\n",
"for it_pipe in range(1,nt+1):\n",
"\n",
" turbine_in.update_LA(LA_soll_vec2[it_pipe])\n",
" turbine_in.set_pressure(Tur_p_nenn)\n",
" Q_in_vec[it_pipe] = turbine_in.get_current_Q()\n",
" reservoir.set_influx(Q_in_vec[it_pipe])\n",
"\n",
"# for each pipeline timestep, execute nt_eRK4 timesteps of the reservoir code\n",
" # set initial condition for the reservoir Con_T_ime evolution calculted with e-RK4\n",
" reservoir.set_pressure(p_old[0],display_warning=False)\n",
" reservoir.set_outflux(Q_old[0],display_warning=False)\n",
" # calculate the Con_T_ime evolution of the reservoir level within each pipeline timestep to avoid runaway numerical error\n",
" for it_res in range(Res_nt):\n",
" reservoir.timestep_reservoir_evolution() \n",
" level_vec[it_pipe] = reservoir.get_current_level() \n",
" volume_vec[it_pipe] = reservoir.get_current_volume() \n",
"\n",
" # get the control variable\n",
" level_control.update_control_variable(level_vec[it_pipe])\n",
" LA_soll_vec[it_pipe] = level_control.get_current_control_variable()\n",
" \n",
" # change the Leitapparatöffnung based on the target value\n",
" turbine.update_LA(LA_soll_vec[it_pipe])\n",
" LA_ist_vec[it_pipe] = turbine.get_current_LA()\n",
"\n",
" # set boundary condition for the next timestep of the characterisCon_T_ic method\n",
" turbine.set_pressure(p_old[-1])\n",
" convergence_parameters[0] = p_old[-2]\n",
" convergence_parameters[1] = v_old[-2]\n",
" turbine.converge(convergence_parameters)\n",
" p_boundary_res[it_pipe] = reservoir.get_current_pressure()\n",
" v_boundary_tur[it_pipe] = 1/Pip_area*turbine.get_current_Q()\n",
" Q_boundary_tur[it_pipe] = turbine.get_current_Q()\n",
"\n",
" # the the boundary condition 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",
" pipe.v[0] = (0.8*pipe.v[0]+0.2*reservoir.get_current_outflux()/Res_area_out)\n",
" p_boundary_tur[it_pipe] = pipe.get_current_pressure_distribution()[-1]\n",
" v_boundary_res[it_pipe] = pipe.get_current_velocity_distribution()[0]\n",
" Q_boundary_res[it_pipe] = pipe.get_current_flux_distribution()[0]\n",
"\n",
" # perform the next timestep via the characterisCon_T_ic method\n",
" pipe.timestep_characteristic_method()\n",
"\n",
" # prepare for next loop\n",
" p_old = pipe.get_current_pressure_distribution()\n",
" v_old = pipe.get_current_velocity_distribution()\n",
" Q_old = pipe.get_current_flux_distribution()\n",
"\n",
"\n",
" # plot some stuff\n",
" # remove line-objects to autoscale axes (there is definetly a better way, but this works ¯\\_(ツ)_/¯ )\n",
" lo_p.remove()\n",
" lo_pmin.remove()\n",
" lo_pmax.remove()\n",
" lo_q.remove()\n",
" lo_qmin.remove()\n",
" lo_qmax.remove()\n",
" # plot new pressure and velocity distribution in the pipeline\n",
" lo_p, = axs1[0].plot(Pip_x_vec,pipe.get_current_pressure_distribution(disp=True),marker='.',c='blue')\n",
" lo_pmin, = axs1[0].plot(Pip_x_vec,pipe.get_lowest_pressure_per_node(disp=True),c='red')\n",
" lo_pmax, = axs1[0].plot(Pip_x_vec,pipe.get_highest_pressure_per_node(disp=True),c='red')\n",
" lo_q, = axs1[1].plot(Pip_x_vec,pipe.get_current_flux_distribution(),marker='.',c='blue')\n",
" lo_qmin, = axs1[1].plot(Pip_x_vec,pipe.get_lowest_flux_per_node(),c='red')\n",
" lo_qmax, = axs1[1].plot(Pip_x_vec,pipe.get_highest_flux_per_node(),c='red')\n",
" fig1.suptitle(str(round(t_vec[it_pipe],2))+ ' s / '+str(round(t_vec[-1],2)) + ' s' )\n",
" fig1.canvas.draw()\n",
" fig1.tight_layout()\n",
" fig1.show()\n",
" plt.pause(0.001) "
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"# plot Con_T_ime evolution of boundary pressure and velocity as well as the reservoir level\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('Level and Volume reservoir')\n",
"axs2.plot(t_vec,level_vec,label='level')\n",
"axs2.set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2.set_ylabel(r'$h$ [m]')\n",
"x_twin_00 = axs2.twinx()\n",
"x_twin_00.set_ylabel(r'$V$ [$\\mathrm{m}^3$]')\n",
"x_twin_00.plot(t_vec,volume_vec)\n",
"axs2.legend()\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('LA')\n",
"axs2.plot(t_vec,100*LA_soll_vec,label='Target')\n",
"axs2.plot(t_vec,100*LA_ist_vec,label='Actual')\n",
"axs2.set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2.set_ylabel(r'$LA$ [%]')\n",
"axs2.legend()\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('Pressure reservoir and turbine')\n",
"axs2.plot(t_vec,pressure_conversion(p_boundary_res,pUnit_calc, pUnit_conv),label='Reservoir')\n",
"axs2.plot(t_vec,pressure_conversion(p_boundary_tur,pUnit_calc, pUnit_conv),label='Turbine')\n",
"axs2.set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2.set_ylabel(r'$p$ ['+pUnit_conv+']')\n",
"axs2.legend()\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('Fluxes')\n",
"axs2.plot(t_vec,Q_boundary_res,label='Outflux')\n",
"axs2.plot(t_vec,Q_in_vec,label='Influx')\n",
"axs2.plot(t_vec,Q_boundary_tur,label='Flux Turbine')\n",
"axs2.set_ylim(-2*Tur_Q_nenn,+2*Tur_Q_nenn)\n",
"axs2.set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2.set_ylabel(r'$Q$ [$\\mathrm{m}^3/\\mathrm{s}$]')\n",
"axs2.legend()\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('Min and Max Pressure')\n",
"axs2.plot(Pip_x_vec,pipe.get_lowest_pressure_per_node(disp=True),c='red')\n",
"axs2.plot(Pip_x_vec,pipe.get_highest_pressure_per_node(disp=True),c='red')\n",
"axs2.set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs2.set_ylabel(r'$p$ ['+pUnit_conv+']')\n",
"\n",
"fig2,axs2 = plt.subplots(1,1)\n",
"axs2.set_title('Min and Max Fluxes')\n",
"axs2.plot(Pip_x_vec,pipe.get_lowest_flux_per_node(),c='red')\n",
"axs2.plot(Pip_x_vec,pipe.get_highest_flux_per_node(),c='red')\n",
"axs2.set_xlabel(r'$x$ [$\\mathrm{m}$]')\n",
"axs2.set_ylabel(r'$Q$ [$\\mathrm{m}^3/\\mathrm{s}$]')\n",
"\n",
"# axs2[0,1].legend()\n",
"# axs2[1,0].legend()\n",
"# axs2[1,1].legend()\n",
"# # axs2[2,0].legend()\n",
"# # axs2[2,1].legend()\n",
"\n",
"\n",
"fig2.tight_layout()\n",
"plt.show()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3.8.13 ('Georg_DT_Slot3')",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
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"vscode": {
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"nbformat_minor": 2
}

168
Turbinen/old/convergence_turbine.py Normal file
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import numpy as np
#importing pressure conversion function
import sys
import os
current = os.path.dirname(os.path.realpath(__file__))
parent = os.path.dirname(current)
sys.path.append(parent)
from functions.pressure_conversion import pressure_conversion
class Francis_Turbine_test:
# units
# make sure that units and print units are the same
# units are used to label graphs and print units are used to have a bearable format when using pythons print()
density_unit = r'$\mathrm{kg}/\mathrm{m}^3$'
flux_unit = r'$\mathrm{m}^3/\mathrm{s}$'
LA_unit = '%'
pressure_unit = 'Pa'
time_unit = 's'
velocity_unit = r'$\mathrm{m}/\mathrm{s}$'
volume_unit = r'$\mathrm{m}^3$'
density_unit_print = 'kg/m³'
flux_unit_print = 'm³/s'
LA_unit_print = '%'
pressure_unit_print = 'mWS'
time_unit_print = 's'
velocity_unit_print = 'm/s'
volume_unit_print = 'm³'
g = 9.81 # m/s² gravitational acceleration
# init
def __init__(self, Q_nenn,p_nenn,t_closing=-1.,timestep=-1.):
self.Q_n = Q_nenn # nominal flux
self.p_n = p_nenn # nominal pressure
self.LA_n = 1. # 100% # nominal Leitapparatöffnung
h = pressure_conversion(p_nenn,'Pa','MWs') # nominal pressure in terms of hydraulic head
self.A = Q_nenn/(np.sqrt(2*self.g*h)*0.98) # Ersatzfläche
self.dt = timestep # simulation timestep
self.t_c = t_closing # closing time
self.d_LA_max_dt = 1/t_closing # maximal change of LA per second
# initialize for get_info() - parameters will be converted to display -1 if not overwritten
self.p = pressure_conversion(-1,self.pressure_unit_print,self.pressure_unit)
self.Q = -1.
self.LA = -0.01
# setter
def set_LA(self,LA,display_warning=True):
# set Leitapparatöffnung
self.LA = LA
# warn user, that the .set_LA() method should not be used ot set LA manually
if display_warning == True:
print('Consider using the .update_LA() method instead of setting LA manually')
def set_timestep(self,timestep,display_warning=True):
# set Leitapparatöffnung
self.dt = timestep
# warn user, that the .set_LA() method should not be used ot set LA manually
if display_warning == True:
print('WARNING: You are changing the timestep of the turbine simulation. This has implications on the simulated closing speed!')
def set_pressure(self,pressure):
# set pressure in front of the turbine
self.p = pressure
#getter
def get_current_Q(self):
# return the flux through the turbine, based on the current pressure in front
# of the turbine and the Leitapparatöffnung
if self.p < 0:
self.Q = 0
else:
self.Q = self.Q_n*(self.LA/self.LA_n)*np.sqrt(self.p/self.p_n)
return self.Q
def get_current_pressure(self):
return self.p
def get_current_LA(self):
return self.LA
def get_info(self, full = False):
new_line = '\n'
p = pressure_conversion(self.p,self.pressure_unit,self.pressure_unit_print)
p_n = pressure_conversion(self.p_n,self.pressure_unit,self.pressure_unit_print)
if full == True:
# :<10 pads the self.value to be 10 characters wide
print_str = (f"Turbine has the following attributes: {new_line}"
f"----------------------------- {new_line}"
f"Type = Francis {new_line}"
f"Nominal flux = {self.Q_n:<10} {self.flux_unit_print} {new_line}"
f"Nominal pressure = {round(p_n,3):<10} {self.pressure_unit_print}{new_line}"
f"Nominal LA = {self.LA_n*100:<10} {self.LA_unit_print} {new_line}"
f"Closing time = {self.t_c:<10} {self.time_unit_print} {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_print} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_print} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_print} {new_line}"
f"Simulation timestep = {self.dt:<10} {self.time_unit_print} {new_line}"
f"----------------------------- {new_line}")
else:
# :<10 pads the self.value to be 10 characters wide
print_str = (f"The current attributes are: {new_line}"
f"----------------------------- {new_line}"
f"Current flux = {self.Q:<10} {self.flux_unit_print} {new_line}"
f"Current pipe pressure = {round(p,3):<10} {self.pressure_unit_print} {new_line}"
f"Current LA = {self.LA*100:<10} {self.LA_unit_print} {new_line}"
f"----------------------------- {new_line}")
print(print_str)
# methods
def update_LA(self,LA_soll):
# update the Leitappartöffnung and consider the restrictions of the closing time of the turbine
LA_diff = self.LA-LA_soll # calculate the difference to the target LA
LA_diff_max = self.d_LA_max_dt*self.dt # calculate the maximum change in LA based on the given timestep
LA_diff = np.sign(LA_diff)*np.min(np.abs([LA_diff,LA_diff_max])) # calulate the correct change in LA
LA_new = self.LA-LA_diff
if LA_new < 0.:
LA_new = 0.
elif LA_new > 1.:
LA_new = 1.
self.set_LA(LA_new,display_warning=False)
def set_steady_state(self,ss_flux,ss_pressure):
# calculate and set steady state LA, that allows the flow of ss_flux at ss_pressure through the
# turbine at the steady state LA
ss_LA = self.LA_n*ss_flux/self.Q_n*np.sqrt(self.p_n/ss_pressure)
if ss_LA < 0 or ss_LA > 1:
raise Exception('LA out of range [0;1]')
self.set_LA(ss_LA,display_warning=False)
def converge(self,area_pipe,pressure_s2l_node,velocity_s2l_node,alpha,f_D,dt):
eps = 1e-9
error = 1.
i = 0
p = pressure_s2l_node
v = velocity_s2l_node
rho = 1000
g = self.g
c = 400
D = area_pipe
p_old = self.get_current_pressure()
Q_old = self.get_current_Q()
v_old = Q_old/area_pipe
while error > eps:
self.set_pressure(p_old)
Q_new = self.get_current_Q()
v_new = Q_new/area_pipe
p_new = p-rho*c*(v_old-v)+rho*c*dt*g*np.sin(alpha)-f_D*rho*c*dt/(2*D)*abs(v)*v
error = abs(Q_old-Q_new)
Q_old = Q_new.copy()
p_old = p_new.copy()
v_old = v_new.copy()
i = i+1
if i == 1e6:
print('did not converge')
break
self.Q = Q_new

312
Turbinen/old/turbine_convergence_test.ipynb Normal file
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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from convergence_turbine import Francis_Turbine_test\n",
"\n",
"#importing pressure conversion function\n",
"import sys\n",
"import os\n",
"current = os.path.dirname(os.path.realpath('Main_Programm.ipynb'))\n",
"parent = os.path.dirname(current)\n",
"sys.path.append(parent)\n",
"from functions.pressure_conversion import pressure_conversion\n",
"from Ausgleichsbecken.Ausgleichsbecken_class_file import Ausgleichsbecken_class\n",
"from Druckrohrleitung.Druckrohrleitung_class_file import Druckrohrleitung_class\n",
"from Regler.Regler_class_file import PI_controller_class"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib qt5\n",
"\n",
"#Turbine\n",
"Q_nenn = 0.85 # m³/s\n",
"p_nenn = pressure_conversion(10.6,'bar','Pa')\n",
"closing_time = 90. #s\n",
"\n",
"#define constants pipe\n",
"\n",
"g = 9.81 # gravitational acceleration [m/s²]\n",
"rho = 1000. # density of water [kg/m³]\n",
"\n",
"# define controller constants\n",
"target_level = 8. # m\n",
"Kp = 0.1\n",
"Ti = 1000.\n",
"deadband_range = 0.05 # m\n",
"\n",
"L = 535.+478. # length of pipeline [m]\n",
"D = 0.9 # pipe diameter [m]\n",
"h_res = target_level # water level in upstream reservoir [m]\n",
"n = 50 # number of pipe segments in discretization\n",
"nt = 10000 # number of time steps after initial conditions\n",
"f_D = 0.014 # 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",
"\n",
"# define constants reservoir\n",
"conversion_pressure_unit = 'mWS'\n",
"\n",
"# preparing the discretization and initial conditions\n",
"initial_flux = Q_nenn/1.1 # 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",
"\n",
"\n",
"# define constants reservoir\n",
"conversion_pressure_unit = 'mWS'\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 = 100 # number of simulation steps of reservoir in between timesteps of pipeline \n",
"simulation_timestep = dt/nt_eRK4"
]
},
{
"cell_type": "code",
"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",
"\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)\n",
"\n",
"\n",
"initial_pressure_turbine = pipe.get_current_pressure_distribution()[-1]\n",
"T1 = Francis_Turbine_test(Q_nenn,p_nenn,closing_time,timestep=dt)\n",
"T1.set_steady_state(initial_flux,initial_pressure_turbine)\n",
"\n",
"T_in = Francis_Turbine_test(Q_nenn,p_nenn,closing_time/2,timestep=dt)\n",
"T_in.set_steady_state(initial_flux,p_nenn)\n",
"\n",
"Pegelregler = PI_controller_class(target_level,deadband_range,Kp,Ti,dt)\n",
"Pegelregler.control_variable = T1.get_current_LA()"
]
},
{
"cell_type": "code",
"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",
"\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",
"p_old = pipe.get_current_pressure_distribution()\n",
"\n",
"# prepare the vectors in which the temporal evolution of the boundary conditions are stored\n",
" # keep in mind, that the velocity at the turbine and the pressure at the reservoir are set manually and\n",
" # through the time evolution of the reservoir respectively \n",
" # the pressure at the turbine and the velocity at the reservoir are calculated from the method of characteristics\n",
"v_boundary_res = np.zeros_like(t_vec)\n",
"v_boundary_tur = np.zeros_like(t_vec)\n",
"p_boundary_res = np.zeros_like(t_vec)\n",
"p_boundary_tur = np.zeros_like(t_vec)\n",
"\n",
"# set the boundary conditions for the first timestep\n",
"v_boundary_res[0] = v_old[0]\n",
"v_boundary_tur[0] = v_old[-1] \n",
"p_boundary_res[0] = p_old[0]\n",
"p_boundary_tur[0] = p_old[-1]\n",
"\n",
"LA_soll_vec = np.full_like(t_vec,T1.get_current_LA())\n",
"LA_ist_vec = np.full_like(t_vec,T1.get_current_LA())\n",
"\n",
"LA_soll_vec2 = np.full_like(t_vec,T_in.get_current_LA())\n",
"LA_soll_vec2[500:1000] = 0.\n",
"LA_soll_vec2[1000:1500] = 1. \n",
"LA_soll_vec2[1500:2000] = 0.\n",
"LA_soll_vec2[2000:2500] = 0.5 "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"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",
"\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",
"\n",
"axs1[0].autoscale()\n",
"axs1[1].autoscale()\n",
"\n",
"fig1.tight_layout()\n",
"plt.pause(1)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"\n",
"for it_pipe in range(1,nt):\n",
"# for each pipeline timestep, execute nt_eRK4 timesteps of the reservoir code\n",
" \n",
" T_in.update_LA(LA_soll_vec2[it_pipe])\n",
" T_in.set_pressure(p_nenn)\n",
" V.set_influx(T_in.get_current_Q())\n",
"\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",
" # 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",
"\n",
" # get the control variable\n",
" Pegelregler.update_control_variable(level_vec[it_pipe])\n",
" LA_soll_vec[it_pipe] = Pegelregler.get_current_control_variable()\n",
" \n",
" # change the Leitapparatöffnung based on the target value\n",
" T1.update_LA(LA_soll_vec[it_pipe])\n",
" LA_ist_vec[it_pipe] = T1.get_current_LA()\n",
"\n",
" # set boundary conditions for the next timestep of the characteristic method\n",
" T1.set_pressure(p_old[-1])\n",
" T1.converge(area_pipe,p_old[-2],v_old[-2],alpha,f_D,dt)\n",
" p_boundary_res[it_pipe] = V.get_current_pressure()\n",
" v_boundary_tur[it_pipe] = T1.get_current_Q()/area_pipe\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",
" pipe.v[0] = (0.8*pipe.v[0]+0.2*V.get_current_outflux()/area_pipe)\n",
" p_boundary_tur[it_pipe] = pipe.get_current_pressure_distribution()[-1]\n",
" v_boundary_res[it_pipe] = pipe.get_current_velocity_distribution()[0]\n",
"\n",
" # perform the next timestep via the characteristic method\n",
" pipe.timestep_characteristic_method()\n",
"\n",
" # prepare for next loop\n",
" p_old = pipe.get_current_pressure_distribution()\n",
" v_old = pipe.get_current_velocity_distribution()\n",
"\n",
" # plot some stuff\n",
" # remove line-objects to autoscale axes (there is definetly a better way, but this works ¯\\_(ツ)_/¯ )\n",
" lo_00.remove()\n",
" 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",
" \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)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"# plot time evolution of boundary pressure and velocity as well as the reservoir level\n",
"\n",
"fig2,axs2 = plt.subplots(3,2)\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].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[0,0].set_ylabel(r'$p$ ['+conversion_pressure_unit+']')\n",
"\n",
"axs2[0,1].set_title('Velocity reservoir')\n",
"axs2[0,1].plot(t_vec,v_boundary_res)\n",
"axs2[0,1].set_ylim(-2*Q_nenn,+2*Q_nenn)\n",
"axs2[0,1].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[0,1].set_ylabel(r'$v$ [$\\mathrm{m}/\\mathrm{s}$]')\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].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[1,0].set_ylabel(r'$p$ ['+conversion_pressure_unit+']')\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",
"\n",
"axs2[2,0].set_title('Level reservoir')\n",
"axs2[2,0].plot(t_vec,level_vec)\n",
"axs2[2,0].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[2,0].set_ylabel(r'$h$ [m]')\n",
"\n",
"axs2[2,1].set_title('LA')\n",
"axs2[2,1].plot(t_vec,100*LA_soll_vec)\n",
"axs2[2,1].plot(t_vec,100*LA_ist_vec)\n",
"axs2[2,1].set_xlabel(r'$t$ [$\\mathrm{s}$]')\n",
"axs2[2,1].set_ylabel(r'$LA$ [%]')\n",
"fig2.tight_layout()\n",
"plt.show()"
]
}
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