# Transmission Constraints¶

Transmission Capacity Rule: The constraint transmission capacity rule defines the variable total transmission capacity $$\kappa_{yaf}$$. The variable total transmission capacity is defined by the constraint as the sum of the parameter transmission capacity installed $$K_{yaf}$$ and the variable new transmission capacity $$\hat{\kappa}_{yaf}$$. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission capacity rule is defined and calculated by the following code fragment:

m.def_transmission_capacity = pyomo.Constraint(
m.tra_tuples,
rule=def_transmission_capacity_rule,
doc='total transmission capacity = inst-cap + new capacity')

def def_transmission_capacity_rule(m, stf, sin, sout, tra, com):
if m.mode['int']:
if (sin, sout, tra, com, stf) in m.inst_tra_tuples:
if (min(m.stf), sin, sout, tra, com) in m.tra_const_cap_dict:
cap_tra = m.transmission_dict['inst-cap'][
(min(m.stf), sin, sout, tra, com)]
else:
cap_tra = (
sum(m.cap_tra_new[stf_built, sin, sout, tra, com]
for stf_built in m.stf
if (sin, sout, tra, com, stf_built, stf) in
m.operational_tra_tuples) +
m.transmission_dict['inst-cap']
[(min(m.stf), sin, sout, tra, com)])
else:
cap_tra = (
sum(m.cap_tra_new[stf_built, sin, sout, tra, com]
for stf_built in m.stf
if (sin, sout, tra, com, stf_built, stf) in
m.operational_tra_tuples))
else:
if (stf, sin, sout, tra, com) in m.tra_const_cap_dict:
cap_tra = \
m.transmission_dict['inst-cap'][(stf, sin, sout, tra, com)]
else:
cap_tra = (m.cap_tra_new[stf, sin, sout, tra, com] +
m.transmission_dict['inst-cap'][
(stf, sin, sout, tra, com)])

return cap_tra


Transmission Output Rule: The constraint transmission output rule defines the variable transmission output commodity flow $$\pi_{yaft}^\text{out}$$. The variable transmission output commodity flow is defined by the constraint as the product of the variable transmission input commodity flow $$\pi_{yaft}^\text{in}$$ and the parameter transmission efficiency $$e_{yaf}$$. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission output rule is defined and calculated by the following code fragment:

m.def_transmission_output = pyomo.Constraint(
m.tm, m.tra_tuples,
rule=def_transmission_output_rule,
doc='transmission output = transmission input * efficiency')

def def_transmission_output_rule(m, tm, stf, sin, sout, tra, com):
return (m.e_tra_out[tm, stf, sin, sout, tra, com] ==
m.e_tra_in[tm, stf, sin, sout, tra, com] *
m.transmission_dict['eff'][(stf, sin, sout, tra, com)])


Transmission Input by Capacity Rule: The constraint transmission input by capacity rule limits the variable transmission input commodity flow $$\pi_{yaft}^\text{in}$$. This constraint prevents the transmission power from exceeding the possible power input capacity of the line. The constraint states that the variable transmission input commodity flow $$\pi_{yaft}^\text{in}$$ must be less than or equal to the variable total transmission capacity $$\kappa_{yaf}$$, scaled by the size of the timesteps $$\Delta t$$. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission input by capacity rule is defined and calculated by the following code fragment:

m.res_transmission_input_by_capacity = pyomo.Constraint(
m.tm, m.tra_tuples,
rule=res_transmission_input_by_capacity_rule,
doc='transmission input <= total transmission capacity')

def res_transmission_input_by_capacity_rule(m, tm, stf, sin, sout, tra, com):
return (m.e_tra_in[tm, stf, sin, sout, tra, com] <=
m.dt * m.cap_tra[stf, sin, sout, tra, com])


Transmission Capacity Limit Rule: The constraint transmission capacity limit rule limits the variable total transmission capacity $$\kappa_{yaf}$$. This constraint restricts a transmission $$f$$ through an arc $$a$$ in support timeframe $$y$$ from having more total power output capacity than an upper bound and having less than a lower bound. The constraint states that the variable total transmission capacity $$\kappa_{yaf}$$ must be greater than or equal to the parameter transmission capacity lower bound $$\underline{K}_{yaf}$$ and less than or equal to the parameter transmission capacity upper bound $$\overline{K}_{yaf}$$. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission capacity limit rule is defined and calculated by the following code fragment:

m.res_transmission_capacity = pyomo.Constraint(
m.tra_tuples,
rule=res_transmission_capacity_rule,
doc='transmission.cap-lo <= total transmission capacity <= '
'transmission.cap-up')

def res_transmission_capacity_rule(m, stf, sin, sout, tra, com):
return (m.transmission_dict['cap-lo'][(stf, sin, sout, tra, com)],
m.cap_tra[stf, sin, sout, tra, com],
m.transmission_dict['cap-up'][(stf, sin, sout, tra, com)])


Transmission Symmetry Rule: The constraint transmission symmetry rule defines the power capacities of incoming and outgoing arcs $$a , a'$$ of a transmission $$f$$ in support timeframe $$y$$. The constraint states that the power capacities $$\kappa_{af}$$ of the incoming arc $$a$$ and the complementary outgoing arc $$a'$$ between two sites must be equal. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission symmetry rule is defined and calculated by the following code fragment:

m.res_transmission_symmetry = pyomo.Constraint(
m.tra_tuples,
rule=res_transmission_symmetry_rule,
doc='total transmission capacity must be symmetric in both directions')

def res_transmission_symmetry_rule(m, stf, sin, sout, tra, com):
return m.cap_tra[stf, sin, sout, tra, com] == (m.cap_tra
[stf, sout, sin, tra, com])


## DCPF Transmission Constraints¶

The following constraints are included in the model if the optional DC Power Flow feature is activated.

DC Power Flow Rule: The constraint DC Power Flow rule defines the power flow of transmission lines, which are modelled with DCPF. This constraint states that the power flow on a transmission line is equal to the product of voltage angle differences of two connecting sites $$v_\text{out}$$ and $${v_\text{in}}$$ and the admittance of the transmission line. This constraint is only applied to the transmission lines modelled with DCPF. The mathematical explanation of this rule is given in Multinode Optimization Model. In script transmission.py the constraint DC Power Flow Rule is defined and calculated by the following code fragment:

m.def_dc_power_flow = pyomo.Constraint(
m.tm, m.tra_tuples_dc,
rule=def_dc_power_flow_rule,
doc='transmission output = (angle(in)-angle(out))/ 57.2958 '
'* -1 *(-1/reactance) * (base voltage)^2')

def def_dc_power_flow_rule(m, tm, stf, sin, sout, tra, com):
return (m.e_tra_in[tm, stf, sin, sout, tra, com] ==
(m.voltage_angle[tm, stf, sin] - m.voltage_angle[tm, stf, sout]) / 57.2958 * -1 *
(-1 / m.transmission_dict['reactance'][(stf, sin, sout, tra, com)])
* m.transmission_dict['base_voltage'][(stf, sin, sout, tra, com)]
* m.transmission_dict['base_voltage'][(stf, sin, sout, tra, com)])


DCPF Transmission Input By Capacity Rule: The constraint DCPF transmission input by capacity rule expands the constraint transmission input by capacity rule for transmission lines modelled with DCPF. This constraint limits the variable transmission input commodity flow $$\pi_{yaft}^\text{in}$$ of DCPF transmission lines also with a lower bound. This constraint prevents the absolute value of the transmission power from exceeding the possible power input capacity of the line especially when the transmission power can be negative. The constraint states that the additive inverse of variable transmission input commodity flow $$-\pi_{yaft}^\text{in}$$ must be less than or equal to the variable total transmission capacity $$\kappa_{yaf}$$, scaled by the size of the time steps :math: Delta t. This constraint is only applied to the tranmission lines modelled with DCPF. The mathematical explanation of this rule is given in Multinode Optimization Model.

In script transmission.py the constraint transmission input by capacity rule is defined and calculated by the following code fragment:

m.res_transmission_dc_input_by_capacity = pyomo.Constraint(
m.tm, m.tra_tuples_dc,
rule=res_transmission_dc_input_by_capacity_rule,
doc='-dcpf transmission input <= total transmission capacity')

def res_transmission_dc_input_by_capacity_rule(m, tm, stf, sin, sout, tra, com):
return (- m.e_tra_in[tm, stf, sin, sout, tra, com] <=
m.dt * m.cap_tra[stf, sin, sout, tra, com])


Voltage Angle Limit Rule: The constraint voltage angle limit rule limits the maximum and minimum difference of voltage angles $$\theta_{yvt}$$ of two sites $$v_\text{out}$$ and $${v_\text{in}}$$ connected with a DCPF transmission line with the parameter voltage angle difference limit $$\overline{dl}_{yaf}$$. This constraint is only applied to the transmission lines modelled with DCPF. The mathematical explanation of this rule is given in Multinode Optimization Model. In script transmission.py the constraint voltage angle limit rule is defined and given by the following code fragment:

m.def_angle_limit = pyomo.Constraint(
m.tm, m.tra_tuples_dc,
rule=def_angle_limit_rule,
doc='-angle limit < angle(in) - angle(out) < angle limit')

def def_angle_limit_rule(m, tm, stf, sin, sout, tra, com):
return (- m.transmission_dict['difflimit'][(stf, sin, sout, tra, com)],
(m.voltage_angle[tm, stf, sin] - m.voltage_angle[tm, stf, sout]),
m.transmission_dict['difflimit'][(stf, sin, sout, tra, com)])


Absolute Transmission Flow Constraints: The two absolute transmission flow constraints are included in the model to create the variable absolute value of transmission commodity flow $${\pi_{yaft}^{\text{in}}}^\prime$$. By limiting the negative $$-{\pi_{yaft}^{\text{in}}}^\prime$$ and positive $${\pi_{yaft}^{\text{in}}}^\prime$$ of substitute variable ‘’e_tra_abs’’ with the variable $$\pi_{yaft}^\text{in}$$ and minimizing the substitute value $${\pi_{yaft}^{\text{in}}}^\prime$$ the absolute value of transmission commodity flow is retrieved. These constraints are only applied to the transmission lines modelled with DCPF. The mathematical explanation of these rules are given in Multinode Optimization Model. In script transmission.py the constraint Absolute Transmission Flow Constraints are defined and given by the following code fragment:

m.e_tra_abs1 = pyomo.Constraint(
m.tm, m.tra_tuples_dc,
rule=e_tra_abs_rule1,
doc='transmission dc input <= absolute transmission dc input')
m.e_tra_abs2 = pyomo.Constraint(
m.tm, m.tra_tuples_dc,
rule=e_tra_abs_rule2,
doc='-transmission dc input <= absolute transmission dc input')

def e_tra_abs_rule1(m, tm, stf, sin, sout, tra, com):
return (m.e_tra_in[tm, stf, sin, sout, tra, com] <=
m.e_tra_abs[tm, stf, sin, sout, tra, com])

def e_tra_abs_rule2(m, tm, stf, sin, sout, tra, com):
return (-m.e_tra_in[tm, stf, sin, sout, tra, com] <=
m.e_tra_abs[tm, stf, sin, sout, tra, com])


Transmission Symmetry Rule: The above mentioned constraint transmission symmetry rule is only applied to the transmission lines modelled with transport model if the DCPF is activated. Since the DCPF transmission lines do not include the complementary arcs, this constraint is ignored for these transmission lines. For this reason, the constraint is indexed with the transmission tuple set m.tra_tuples_tp if the DCPF is activated.

In script transmission.py the constraint transmission symmetry rule is defined as following if the DCPF is activated:

m.res_transmission_symmetry = pyomo.Constraint(
m.tra_tuples_tp,
rule=res_transmission_symmetry_rule,
doc='total transmission capacity must be symmetric in both directions')