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define_model.py
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define_model.py
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# -*- coding: utf-8 -*-
from gekko import GEKKO
import datetime
def define_model(config):
# Select server
server = config.server
# Application name
app = 'hale_' + '{:%Y_%m_%d_%H_%M_%S}'.format(datetime.datetime.now())
#Initialize model
m = GEKKO(remote=True,server=server,name=app)
#%% Constants
pi = 3.141592653
# Atmospheric Effects
g = 9.80665 # Gravity (m/s^2)
R_air = 287.041 # Gas Constant for air (m^2/(s^2 K))
rho_11 = 0.364 # Standard air density at 11 km (kg/m^3)
T_11 = 216.66 # Standard air temp at 11 km (K)
# Flight Dynamics
mass = config.aircraft.mass_total.value # Total Mass (kg)
E_d = config.aircraft.battery_energy_density.value # Battery energy density (W*hr/kg)
m_battery = config.aircraft.mass_battery.value # (kg)
max_dist = config.distance.max # Maximum orbit radius (m)
# Drag Model (Translated from Judd's Matlab code)
S = config.aircraft.wing_top_surface_area.value # Wing and solar panel area (m^2)
chord = config.aircraft.chord.value
# Propeller Efficiency
R_prop = config.aircraft.propeller_radius.value # Propeller Radius (m) - Kevin
# Power
e_motor = config.aircraft.motor_efficiency.value # Efficiency of motor
P_payload = config.aircraft.power_for_payload.value # Power for payload (W)
E_batmax = m_battery*E_d*3.6/1000 #m.Const(m_battery.value*E_d.value*3.6/1000,'E_batmax') # Max energy stored in battery (MJ)
# Initial Conditions (from final initialization)
h_0 = config.h.initial_value
if(config.use_wind):
v_a0 = config.aircraft.v.initial_value
v_g0 = config.aircraft.v.initial_value
else:
v_0 = config.aircraft.v.initial_value
gamma_0 = config.aircraft.gamma.level # Initial flight path angle (rad)
alpha_0 = config.aircraft.alpha.initial_value # Initial angle of attack (rad)
psi_0 = config.aircraft.psi.initial_value # Initial heading (rad)
phi_0 = config.aircraft.phi.initial_value # Initial bank angle (rad)
tp_0 = config.aircraft.tp.initial_value # Initial thrust
SOC_initial = config.aircraft.battery_initial_SOC.value # Initial state of charge
E_Batt_0 = E_batmax*SOC_initial # Initial Battery Charge
#%% Parameters
t = m.Param(name='t',value=0) # Time
flux = m.Param(name='flux',value=0) # Direct tracking flux from SMARTS
sunset = m.Param(name='sunset',value=0) # Time step at which sun sets
zenith = m.Param(name='zenith',value=0) # Solar zenith from SMARTS (0=up, 90=horizon)
azimuth = m.Param(name='azimuth',value=0) # Solar azimuth from SMARTS (clockwise from north)
sn1 = m.Param(name='sn1',value=0) # Normalized sun direction vector
sn2 = m.Param(name='sn2',value=0) # Normalized sun direction vector
sn3 = m.Param(name='sn3',value=0) # Normalized sun direction vector
sun_h = m.Param(name='sun_h',value=0) # Flux on horizontal surface
# FVs and MVs need to be parameters
tp = m.MV(name='tp',value=tp_0) #3.171
alpha = m.MV(name='alpha',value=alpha_0) # Angle of Attack (rad)
phi = m.MV(name='phi',value=phi_0) #0.038
p_bat = m.MV(name='p_bat',value=0) # Power used to charge or discharge battery (W)
if(config.use_wind):
# Wind
w_n = m.Param(name='w_n',value=config.w_n)
w_e = m.Param(name='w_e',value=config.w_e)
w_d = m.Param(name='w_d',value=config.w_d)
#%% Variables
# Atmospheric Effects
h = m.Var(name='h',value=h_0,lb=config.h.min,ub=config.h.max) # Height from sea level (m) (27432 m = 90,000 ft, 18288 m = 60,000 ft)
# Flight Dynamics
if(config.use_wind):
v_a = m.Var(name='v_a',value=v_a0,lb=config.aircraft.v.min) # Velocity (m/s)
v_g = m.Var(name='v_g',value=v_g0) # Velocity (m/s)
chi = m.Var(name='chi',value=psi_0) # Heading angle
else:
v = m.Var(name='v',value=v_0,lb=config.aircraft.v.min) # Velocity (m/s)
psi = m.Var(name='psi',value=psi_0) # Heading angle
gamma = m.Var(name='gamma',value=gamma_0,lb=config.aircraft.gamma.min,ub=config.aircraft.gamma.max) # Flight path angle (rad)
x = m.Var(name='x',value=config.x.initial_value) # Horizontal distance (m)
y = m.Var(name='y',value=config.y.initial_value) # Other horizontal distance
dist = m.CV(name='dist',value=m.sqrt(x**2+y**2),lb=0,ub=max_dist*1.1)
# Solar
mu_clipped = m.Var(name='mu_clipped',value=0,lb=0)
mu_slack = m.Var(name='mu_slack',value=0,lb=0)
# Power
e_batt = m.Var(name='e_batt',value=E_Batt_0,ub=E_batmax) # Energy stored in battery (MJ)
te = m.Var(name='te',value=E_Batt_0+mass*g*(h-h_0)*1e-6) # Total energy (MJ)
p_total = m.Var(name='p_total',value=0,lb=0) # Energy balance
#%% Intermediates
#### Atmospheric Effects
rho = m.Intermediate(rho_11*m.exp(-(g/(R_air*T_11))*(h-11000)),'rho') # Air density
T_air = m.Intermediate(-56.46+273.15,'T_air') # Air temperature (isothermal region)
mu = m.Intermediate(1.458e-6*m.sqrt(T_air)/(1+110.4/T_air),'mu') # Dynamic viscosity of air (kg/(m s))
# Pitch from AoA
theta = m.Intermediate(gamma+alpha,'theta')
if(config.use_wind):
v_w = m.Intermediate(m.sqrt(w_n**2+w_e**2+w_d**2),'v_w')
gamma_a = m.Intermediate((v_g*gamma + w_d)/v_a,'gamma_a')
psi = m.Intermediate(chi - m.asin((-w_n*m.sin(chi) + w_e*m.cos(chi))/(v_a*m.cos(gamma_a))),'psi')
dx = m.Intermediate(x.dt(),'dx')
dy = m.Intermediate(y.dt(),'dy')
psi_n = m.Intermediate(m.cos(psi),'psi_n')
psi_e = m.Intermediate(m.sin(psi),'psi_e')
v_a_n = m.Intermediate(dx-w_n,'v_a_n')
v_a_e = m.Intermediate(dy-w_e,'v_a_e')
beta = m.Intermediate(m.acos((v_a_n * psi_n + v_a_e * psi_e)/(m.sqrt(v_a_n**2+v_a_e**2)*m.sqrt(psi_n**2+psi_e**2))),'beta')
# Flat plate Reynolds number
if(config.use_wind):
Re = m.Intermediate(rho*v_a*chord/mu,'Re') # Reynolds number
else:
Re = m.Intermediate(rho*v*chord/mu,'Re') # Reynolds number
# Get alpha in degrees for fitted functions
alpha_deg = alpha*180/pi
# Wing lift coefficient
cl = m.Intermediate(config.aircraft.CL(alpha_deg,Re),'cl')
# Wing drag coefficient
C_D = m.Intermediate(config.aircraft.CD(alpha_deg,Re),'C_D')
#### Flight Dynamics
if(config.use_wind):
q = m.Intermediate(1/2*rho*v_a**2,'q') # Dynamic pressure (Pa)
else:
q = m.Intermediate(1/2*rho*v**2,'q') # Dynamic pressure (Pa)
L = m.Intermediate(q*cl*S,'L') # Lift (N)
D = m.Intermediate(C_D*q*S,'D') # Drag (N)
nh = m.Intermediate(L*m.sin(phi)/(mass*g),'nh') # Horizontal load factor
nv = m.Intermediate(L*m.cos(phi)/(mass*g),'nv') # Vertical load factor
### Propeller Max Theoretical Efficiency
Adisk = m.Intermediate(pi*R_prop**2,'Adisk') # Area of disk
if(config.use_wind):
e_prop = m.Intermediate(2.0/(1.0+(tp/(Adisk*v_a**2.0*rho/2.0)+1.0)**0.5),'e_prop')
else:
e_prop = m.Intermediate(2.0/(1.0+(tp/(Adisk*v**2.0*rho/2.0)+1.0)**0.5),'e_prop')
nu_prop = m.Intermediate(e_prop*e_motor,'nu_prop')
### Power
if(config.use_wind):
P_N = m.Intermediate(P_payload+v_a*tp/nu_prop,'P_N') # Power Needed by Aircraft
else:
P_N = m.Intermediate(P_payload+v*tp/nu_prop,'P_N') # Power Needed by Aircraft
#### Solar (with orientation correction)
c1 = m.Intermediate(m.cos(-phi),'c1')
c2 = m.Intermediate(m.cos(-theta),'c2')
c3 = m.Intermediate(m.cos(psi),'c3')
s1 = m.Intermediate(m.sin(-phi),'s1')
s2 = m.Intermediate(m.sin(-theta),'s2')
s3 = m.Intermediate(m.sin(psi),'s3')
n1 = m.Intermediate(c1*s2*s3-c3*s1,'n1')
n2 = m.Intermediate(c1*c3*s2+s1*s3,'n2')
n3 = m.Intermediate(c1*c2,'n3')
nn = m.Intermediate(m.sqrt(n1**2+n2**2+n3**2),'nn')
mu_solar = m.Intermediate(sn1*n1/nn+sn2*n2/nn+sn3*n3/nn,'mu_solar') # Obliquity factor
G_sol = m.Intermediate(flux*mu_clipped,'G_sol') # Orientation adjusted solar flux (W/m^2)
panel_efficiency = m.Intermediate(config.aircraft.panel_efficiency(G_sol,m),'panel_efficiency') # Solar panel efficiency
P_solar = m.Intermediate(mu_clipped*panel_efficiency*S*flux,'P_solar') # Total power generated by panel (W)
#%% Equations
# Flight Dynamics
if(config.use_wind):
m.Equation(v_g.dt()==((tp-D)/(mass*g)-m.sin(gamma))*g)
m.Equation(gamma.dt()*v_g==g*(nv-m.cos(gamma)))
m.Equation(chi.dt()*v_g*m.cos(gamma)==g*nh*m.cos(chi-psi))
m.Equation(h.dt()==v_g*m.sin(gamma))
m.Equation(x.dt()==v_g*m.cos(chi)*m.cos(gamma))
m.Equation(y.dt()==v_g*m.sin(chi)*m.cos(gamma))
m.Equation(dist==m.sqrt(x**2+y**2))
m.Equation(v_a == m.sqrt(v_g**2-2*v_g*(w_n*m.cos(chi)*m.cos(gamma)+w_e*m.sin(chi)*m.cos(gamma)-w_d*m.sin(gamma))+v_w**2))
else:
m.Equation(v.dt()==((tp-D)/(mass*g)-m.sin(gamma))*g)
m.Equation(gamma.dt()*v==g*(nv-m.cos(gamma)))
m.Equation(psi.dt()*v*m.cos(gamma)==g*nh)
m.Equation(h.dt()==v*m.sin(gamma))
m.Equation(x.dt()==v*m.cos(psi)*m.cos(gamma))
m.Equation(y.dt()==v*m.sin(psi)*m.cos(gamma))
m.Equation(dist==m.sqrt(x**2+y**2))
# Power
m.Equation(e_batt.dt()==p_bat*1e-6) # (Convert to MJ) p_bat is the charging rate of the battery.
m.Equation(p_total==P_solar-p_bat-P_N)
m.Equation(te==e_batt+mass*g*(h-h_0)*1e-6)
# Solar
# This clips mu to be greater than 0
m.Equation(mu_clipped==mu_solar+mu_slack)
m.Equation(mu_clipped*mu_slack<=1e-4)
# Objective - Maximize total energy
m.Obj(-te)
#%% End Model
m.t=t
m.flux=flux
m.sunset=sunset
m.zenith=zenith
m.azimuth=azimuth
m.sn1=sn1
m.sn2=sn2
m.sn3=sn3
m.sun_h=sun_h
m.tp=tp
m.alpha=alpha
m.phi=phi
m.p_bat=p_bat
m.h=h
m.gamma=gamma
m.psi=psi
m.x=x
m.y=y
m.dist=dist
m.cl=cl
m.mu_clipped=mu_clipped
m.mu_slack=mu_slack
m.e_batt=e_batt
m.te=te
m.p_total=p_total
if(config.use_wind):
m.chi = chi
m.v_a = v_a
m.v_g = v_g
m.beta = beta
else:
m.v=v
return m