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Problem Specifications

Optimal Power Flow (OPF)

Objective

objective_min_fuel_cost(pm)

Variables

variable_voltage(pm)
variable_active_generation(pm)
variable_reactive_generation(pm)
variable_branch_flow(pm)
variable_dcline_flow(pm)

Constraints

constraint_model_voltage(pm)
for i in ids(pm, :ref_buses)
    constraint_theta_ref(pm, i)
end
for i in ids(pm, :bus)
    constraint_power_balance_shunt(pm, i)
end
for i in ids(pm, :branch)
    constraint_ohms_yt_from(pm, i)
    constraint_ohms_yt_to(pm, i)

    constraint_voltage_angle_difference(pm, i)

    constraint_thermal_limit_from(pm, i)
    constraint_thermal_limit_to(pm, i)
end
for i in ids(pm, :dcline)
    constraint_dcline(pm, i)
end

Optimal Power Flow (OPF) using the Branch Flow Model

Objective

objective_min_fuel_cost(pm)

Variables

variable_voltage(pm)
variable_active_generation(pm)
variable_reactive_generation(pm)
variable_branch_flow(pm)
variable_branch_current(pm)
variable_dcline_flow(pm)

Constraints

constraint_model_voltage(pm)
for i in ids(pm, :ref_buses)
    constraint_theta_ref(pm, i)
end
for i in ids(pm, :bus)
    constraint_power_balance_shunt(pm, i)
end
for i in ids(pm, :branch)
    constraint_flow_losses(pm, i)
    constraint_voltage_magnitude_difference(pm, i)

    constraint_voltage_angle_difference(pm, i)

    constraint_thermal_limit_from(pm, i)
    constraint_thermal_limit_to(pm, i)
end
for i in ids(pm, :dcline)
    constraint_dcline(pm, i)
end

Optimal Transmission Switching (OTS)

General Assumptions

  • if the branch status is 0 in the input, it is out of service and forced to 0 in OTS

Variables

variable_branch_indicator(pm)
variable_voltage_on_off(pm)
variable_generation(pm)
variable_branch_flow(pm)
variable_dcline_flow(pm)

Objective

objective_min_fuel_cost(pm)

Constraints

constraint_model_voltage_on_off(pm)
for i in ids(pm, :ref_buses)
    constraint_theta_ref(pm, i)
end
for i in ids(pm, :bus)
    constraint_power_balance_shunt(pm, i)
end
for i in ids(pm, :branch)
    constraint_ohms_yt_from_on_off(pm, i)
    constraint_ohms_yt_to_on_off(pm, i)

    constraint_voltage_angle_difference_on_off(pm, i)

    constraint_thermal_limit_from_on_off(pm, i)
    constraint_thermal_limit_to_on_off(pm, i)
end
for i in ids(pm, :dcline)
    constraint_dcline(pm, i)
end

Power Flow (PF)

Assumptions

Variables

variable_voltage(pm, bounded = false)
variable_active_generation(pm, bounded = false)
variable_reactive_generation(pm, bounded = false)
variable_branch_flow(pm, bounded = false)
variable_dcline_flow(pm, bounded = false)

Constraints

constraint_model_voltage(pm)
for (i,bus) in ref(pm, :ref_buses)
    @assert bus["bus_type"] == 3
    constraint_theta_ref(pm, i)
    constraint_voltage_magnitude_setpoint(pm, i)
end
for (i,bus) in ref(pm, :bus)
    constraint_power_balance_shunt(pm, i)
    # PV Bus Constraints
    if length(ref(pm, :bus_gens, i)) > 0 && !(i in ids(pm,:ref_buses))
        # this assumes inactive generators are filtered out of bus_gens
        @assert bus["bus_type"] == 2
        constraint_voltage_magnitude_setpoint(pm, i)
        for j in ref(pm, :bus_gens, i)
            constraint_active_gen_setpoint(pm, j)
        end
    end
end
for i in ids(pm, :branch)
    constraint_ohms_yt_from(pm, i)
    constraint_ohms_yt_to(pm, i)
end
for (i,dcline) in ref(pm, :dcline)
    constraint_active_dcline_setpoint(pm, i)

    f_bus = ref(pm, :bus)[dcline["f_bus"]]
    if f_bus["bus_type"] == 1
        constraint_voltage_magnitude_setpoint(pm, f_bus["index"])
    end

    t_bus = ref(pm, :bus)[dcline["t_bus"]]
    if t_bus["bus_type"] == 1
        constraint_voltage_magnitude_setpoint(pm, t_bus["index"])
    end
end

Power Flow (PF) using the Branch Flow Model

Assumptions

Variables

variable_voltage(pm, bounded = false)
variable_active_generation(pm, bounded = false)
variable_reactive_generation(pm, bounded = false)
variable_branch_flow(pm, bounded = false)
variable_branch_current(pm, bounded = false)
variable_dcline_flow(pm, bounded = false)

Constraints

constraint_model_voltage(pm)
for (i,bus) in ref(pm, :ref_buses)
    @assert bus["bus_type"] == 3
    constraint_theta_ref(pm, i)
    constraint_voltage_magnitude_setpoint(pm, i)
end
for (i,bus) in ref(pm, :bus)
    constraint_power_balance_shunt(pm, i)
    if length(ref(pm, :bus_gens, i)) > 0 && !(i in ids(pm,:ref_buses))
        # this assumes inactive generators are filtered out of bus_gens
        @assert bus["bus_type"] == 2
        constraint_voltage_magnitude_setpoint(pm, i)
        for j in ref(pm, :bus_gens, i)
            constraint_active_gen_setpoint(pm, j)
        end
    end
end
for i in ids(pm, :branch)
    constraint_flow_losses(pm, i)
    constraint_voltage_magnitude_difference(pm, i)
end
for (i,dcline) in ref(pm, :dcline)
    constraint_active_dcline_setpoint(pm, i)

    f_bus = ref(pm, :bus)[dcline["f_bus"]]
    if f_bus["bus_type"] == 1
        constraint_voltage_magnitude_setpoint(pm, f_bus["index"])
    end

    t_bus = ref(pm, :bus)[dcline["t_bus"]]
    if t_bus["bus_type"] == 1
        constraint_voltage_magnitude_setpoint(pm, t_bus["index"])
    end
end

Transmission Network Expansion Planning (TNEP)

Objective

objective_tnep_cost(pm)

Variables

variable_branch_ne(pm)
variable_voltage(pm)
variable_voltage_ne(pm)
variable_active_generation(pm)
variable_reactive_generation(pm)
variable_branch_flow(pm)
variable_dcline_flow(pm)
variable_branch_flow_ne(pm)

Constraints

constraint_model_voltage(pm)
constraint_model_voltage_ne(pm)
for i in ids(pm, :ref_buses)
    constraint_theta_ref(pm, i)
end
for i in ids(pm, :bus)
    constraint_power_balance_shunt_ne(pm, i)
end
for i in ids(pm, :branch)
    constraint_ohms_yt_from(pm, i)
    constraint_ohms_yt_to(pm, i)

    constraint_voltage_angle_difference(pm, i)

    constraint_thermal_limit_from(pm, i)
    constraint_thermal_limit_to(pm, i)
end
for i in ids(pm, :ne_branch)
    constraint_ohms_yt_from_ne(pm, i)
    constraint_ohms_yt_to_ne(pm, i)

    constraint_voltage_angle_difference_ne(pm, i)

    constraint_thermal_limit_from_ne(pm, i)
    constraint_thermal_limit_to_ne(pm, i)
end
for i in ids(pm, :dcline)
    constraint_dcline(pm, i)
end