def optimal_adapt_level(gross_damages, m, r):
if value(m.adapt_g1[r]) * value(m.adapt_g2[r]) == 0:
return 0
eps = 0.0005
return (soft_min(gross_damages, 0.01) / (m.adapt_g1[r] * m.adapt_g2[r]) + eps) ** (
1 / (m.adapt_g2[r] - 1)
)
def damage_fct(temperature, init_temp, m, r):
power_fct = (
lambda temp: m.damage_a1[r] * temp + m.damage_a2[r] * temp ** m.damage_a3[r]
)
damage = power_fct(soft_min(temperature))
if init_temp is not None:
damage -= power_fct(init_temp)
return damage
def calc_utility(consumption, population, elasmu):
return (soft_min(consumption / population)**
(1 - elasmu) - 1) / (1 - elasmu) - 1
def adaptation_costs(adapt_level, m, r):
return m.adapt_g1[r] * soft_min(adapt_level) ** m.adapt_g2[r]
def get_constraints(m: AbstractModel) -> Sequence[GeneralConstraint]:
"""Damage and adaptation costs equations and constraints
(RICE specification)
Necessary variables:
m.damage_costs (sum of residual damages and adaptation costs multiplied by gross GDP)
Returns:
list of constraints (any of:
- GlobalConstraint
- GlobalInitConstraint
- RegionalConstraint
- RegionalInitConstraint
)
"""
constraints = []
m.damage_costs = Var(m.t, m.regions)
m.gross_damages = Var(m.t, m.regions)
m.resid_damages = Var(m.t, m.regions)
m.damage_a1 = Param(m.regions)
m.damage_a2 = Param(m.regions)
m.damage_a3 = Param(m.regions)
m.damage_scale_factor = Param()
m.adapt_level = Var(m.t, m.regions, within=NonNegativeReals)
m.adapt_costs = Var(m.t, m.regions)
m.adapt_FAD = Var(m.t, m.regions, bounds=(0, 0.15), initialize=0)
m.adapt_IAD = Var(m.t, m.regions, bounds=(0, 0.15), initialize=0)
m.adap1 = Param(m.regions)
m.adap2 = Param(m.regions)
m.adap3 = Param(m.regions)
m.adapt_rho = Param()
m.fixed_adaptation = Param()
m.adapt_SAD = Var(m.t, m.regions, initialize=0.01, within=NonNegativeReals)
# SLR damages
m.SLRdam1 = Param(m.regions)
m.SLRdam2 = Param(m.regions)
m.SLR_damages = Var(m.t, m.regions)
constraints.extend(
[
RegionalConstraint(
lambda m, t, r: m.SLR_damages[t, r]
== slr_damages(
m.total_SLR[t],
m.GDP_gross[t, r],
m.GDP_gross[0, r],
m.SLRdam1[r],
m.SLRdam2[r],
),
"SLR_damages",
),
]
)
# Gross damages and adaptation levels
constraints.extend(
[
RegionalConstraint(
lambda m, t, r: m.gross_damages[t, r]
== m.damage_scale_factor * damage_fct(m.temperature[t], m.T0, m, r),
"gross_damages",
),
RegionalConstraint(
lambda m, t, r: m.resid_damages[t, r]
== m.gross_damages[t, r] / (1 + m.adapt_level[t, r])
+ m.SLR_damages[t, r],
"resid_damages",
),
RegionalConstraint(
lambda m, t, r: m.adapt_SAD[t, r]
== (1 - m.dk) ** m.dt * m.adapt_SAD[t - 1, r] + m.adapt_IAD[t, r]
if t > 0
else Constraint.Skip,
"adapt_SAD",
),
RegionalInitConstraint(lambda m, r: m.adapt_SAD[0, r] == 0),
RegionalInitConstraint(lambda m, r: m.adapt_IAD[0, r] == 0),
RegionalInitConstraint(lambda m, r: m.adapt_FAD[0, r] == 0),
RegionalConstraint(
lambda m, t, r: (
m.adapt_level[t, r]
== m.adap1[r]
* (
m.adap2[r]
* soft_min(m.adapt_FAD[t, r], scale=0.005) ** m.adapt_rho
+ (1 - m.adap2[r])
* soft_min(m.adapt_SAD[t, r], scale=0.005) ** m.adapt_rho
)
** (m.adap3[r] / m.adapt_rho)
)
if t > 0
else (m.adapt_level[t, r] == 0),
name="adapt_level",
),
RegionalConstraint(
lambda m, t, r: m.adapt_costs[t, r]
== m.adapt_FAD[t, r] + m.adapt_IAD[t, r],
"adapt_costs",
),
RegionalConstraint(
lambda m, t, r: m.damage_costs[t, r]
== m.resid_damages[t, r] + m.adapt_costs[t, r],
"damage_costs",
),
]
)
return constraints
def slr_damages(total_slr, gdp_t, gdp_0, dam1, dam2):
return (
4 * (dam1 * total_slr + dam2 * total_slr ** 2) * soft_min(gdp_t / gdp_0) ** 0.25
)
def get_constraints(m: AbstractModel) -> Sequence[GeneralConstraint]:
"""Abatement cost equations and constraints
Necessary variables:
m.abatement_costs
Returns:
list of constraints (any of:
- GlobalConstraint
- GlobalInitConstraint
- RegionalConstraint
- RegionalInitConstraint
)
"""
constraints = []
### Technological learning
# Learning by doing
m.LBD_rate = Param()
m.log_LBD_rate = Param(initialize=log(m.LBD_rate) / log(2))
m.LBD_scaling = Param()
m.LBD_factor = Var(m.t) # , bounds=(0,1), initialize=1)
constraints.append(
GlobalConstraint(
lambda m, t: m.LBD_factor[t]
== soft_min(
(
m.baseline_cumulative_global(m, m.year(0), m.year(t))
- m.cumulative_emissions[t]
)
/ m.LBD_scaling
+ 1.0
)
** m.log_LBD_rate,
name="LBD",
)
)
# Learning over time and total learning factor
m.LOT_rate = Param()
m.LOT_factor = Var(m.t)
m.learning_factor = Var(m.t)
constraints.extend(
[
GlobalConstraint(
lambda m, t: m.LOT_factor[t] == 1 / (1 + m.LOT_rate) ** t, "LOT"
),
GlobalConstraint(
lambda m, t: m.learning_factor[t]
== (m.LBD_factor[t] * m.LOT_factor[t]),
"learning",
),
]
)
# Abatement costs and MAC
m.abatement_costs = Var(m.t, m.regions, within=NonNegativeReals, initialize=0)
m.area_under_MAC = Var(m.t, m.regions, within=NonNegativeReals, initialize=0)
m.rel_abatement_costs = Var(m.t, m.regions, bounds=(0, 0.3))
m.MAC_gamma = Param()
m.MAC_beta = Param()
m.MAC_scaling_factor = Param(m.regions) # Regional scaling of the MAC
m.carbonprice = Var(m.t, m.regions, bounds=lambda m: (0, 1.5 * m.MAC_gamma))
constraints.extend(
[
RegionalConstraint(
lambda m, t, r: (
m.area_under_MAC[t, r]
if value(m.allow_trade)
else m.abatement_costs[t, r]
)
== AC(
m.relative_abatement[t, r],
m.learning_factor[t],
m.MAC_gamma,
m.MAC_beta,
m.MAC_scaling_factor[r],
)
* m.baseline[t, r],
"abatement_costs",
),
RegionalConstraint(
lambda m, t, r: m.rel_abatement_costs[t, r]
== m.abatement_costs[t, r] / m.GDP_gross[t, r],
"rel_abatement_costs",
),
RegionalConstraint(
lambda m, t, r: m.carbonprice[t, r]
== MAC(
m.relative_abatement[t, r],
m.learning_factor[t],
m.MAC_gamma,
m.MAC_beta,
m.MAC_scaling_factor[r],
),
"carbonprice",
),
RegionalInitConstraint(
lambda m, r: m.carbonprice[0, r] == 0, "init_carbon_price"
),
]
)
# How are mitigation costs distributed over regions?
m.allow_trade = Param()
constraints.extend(
[
GlobalConstraint(
lambda m, t: sum(m.abatement_costs[t, r] for r in m.regions)
== sum(m.area_under_MAC[t, r] for r in m.regions)
if m.allow_trade
else Constraint.Skip
),
RegionalConstraint(
lambda m, t, r: m.abatement_costs[t, r] <= 1.5 * m.area_under_MAC[t, r]
if m.allow_trade
else Constraint.Skip
),
# RegionalConstraint(
# lambda m, t, r: m.abatement_costs[t, r] >= 0.5 * m.area_under_MAC[t, r]
# if m.allow_trade
# else Constraint.Skip
# ),
]
)
# Keep track of relative global costs
m.global_rel_abatement_costs = Var(m.t)
constraints.extend(
[
GlobalConstraint(
lambda m, t: m.global_rel_abatement_costs[t]
== sum(m.abatement_costs[t, r] for r in m.regions)
/ sum(m.GDP_gross[t, r] for r in m.regions),
"global_rel_abatement_costs",
)
]
)
return constraints
def get_constraints(m: AbstractModel) -> Sequence[GeneralConstraint]:
"""Economics and Cobb-Douglas equations and constraints
Necessary variables:
m.L (equal to m.population)
m.dk
Returns:
list of constraints (any of:
- GlobalConstraint
- GlobalInitConstraint
- RegionalConstraint
- RegionalInitConstraint
)
"""
constraints = []
m.init_capitalstock_factor = Param(m.regions)
m.capital_stock = Var(
m.t,
m.regions,
initialize=lambda m, t, r: m.init_capitalstock_factor[r] * m.GDP(
m.year(t), r),
)
# Parameters
m.alpha = Param()
m.dk = Param()
m.sr = Param()
m.GDP_gross = Var(m.t,
m.regions,
initialize=lambda m, t, r: m.GDP(m.year(0), r))
m.GDP_net = Var(m.t, m.regions)
m.investments = Var(m.t, m.regions)
m.consumption = Var(m.t, m.regions)
m.ignore_damages = Param()
# Cobb-Douglas, GDP, investments, capital and consumption
constraints.extend([
RegionalConstraint(
lambda m, t, r: m.GDP_gross[t, r] == economics.calc_GDP(
m.TFP(m.year(t), r),
m.L(m.year(t), r),
soft_min(m.capital_stock[t, r], scale=10),
m.alpha,
),
"GDP_gross",
),
RegionalConstraint(
lambda m, t, r: m.GDP_net[t, r] == m.GDP_gross[t, r] *
(1 - (m.damage_costs[t, r] if not value(m.ignore_damages) else 0)
) - m.abatement_costs[t, r],
"GDP_net",
),
RegionalConstraint(
lambda m, t, r: m.investments[t, r] == m.sr * m.GDP_net[t, r],
"investments",
),
RegionalConstraint(
lambda m, t, r: m.consumption[t, r] ==
(1 - m.sr) * m.GDP_net[t, r],
"consumption",
),
RegionalConstraint(
lambda m, t, r: (m.capital_stock[t, r] == m.capital_stock[
t - 1, r] + m.dt * economics.calc_dKdt(m.capital_stock[
t, r], m.dk, m.investments[t, r], m.dt))
if t > 0 else Constraint.Skip,
"capital_stock",
),
RegionalInitConstraint(lambda m, r: m.capital_stock[
0, r] == m.init_capitalstock_factor[r] * m.GDP(m.year(0), r)),
])
return constraints