PowerSystems.Branch Formulations

Table of contents

1. StaticBranch
2. StaticBranchBounds
3. StaticBranchUnbounded
4. HVDCTwoTerminalUnbounded
5. HVDCTwoTerminalLossless
6. HVDCTwoTerminalDispatch
7. PhaseAngleControl
8. TwoTerminalLCCLine
9. Valid configurations

StaticBranch

Formulation valid for PTDFPowerModel Network model

Variables:

• FlowActivePowerVariable:

  • Bounds: $(-\infty,\infty)$
  • Symbol: $f$ If Slack variables are enabled:

• FlowActivePowerSlackUpperBound:

  • Bounds: [0.0, ]
  • Default proportional cost: 2e5
  • Symbol: $f^\text{sl,up}$

• FlowActivePowerSlackLowerBound:

  • Bounds: [0.0, ]
  • Default proportional cost: 2e5
  • Symbol: $f^\text{sl,lo}$

Static Parameters

• $R^\text{max}$ = PowerSystems.get_rating(branch)

Objective:

Add a large proportional cost to the objective function if rate constraint slack variables are used $+ (f^\text{sl,up} + f^\text{sl,lo}) \cdot 2 \cdot 10^5$

Expressions:

No expressions are used.

Constraints:

For each branch $b \in \{1,\dots, B\}$ (in a system with $N$ buses) the constraints are given by:

\[\begin{aligned} & f_t = \sum_{i=1}^N \text{PTDF}_{i,b} \cdot \text{Bal}_{i,t}, \quad \forall t \in \{1,\dots, T\}\\ & f_t - f_t^\text{sl,up} \le R^\text{max},\quad \forall t \in \{1,\dots, T\} \\ & f_t + f_t^\text{sl,lo} \ge -R^\text{max},\quad \forall t \in \{1,\dots, T\} \end{aligned}\]

on which $\text{PTDF}$ is the $N \times B$ system Power Transfer Distribution Factors (PTDF) matrix, and $\text{Bal}_{i,t}$ is the active power bus balance expression (i.e. $\text{Generation}_{i,t} - \text{Demand}_{i,t}$) at bus $i$ at time-step $t$.

StaticBranchBounds

Formulation valid for PTDFPowerModel Network model

Variables:

• FlowActivePowerVariable:
  • Bounds: $\left[-R^\text{max},R^\text{max}\right]$
  • Symbol: $f$

Static Parameters

• $R^\text{max}$ = PowerSystems.get_rating(branch)

Objective:

No cost is added to the objective function.

Expressions:

No expressions are used.

Constraints:

For each branch $b \in \{1,\dots, B\}$ (in a system with $N$ buses) the constraints are given by:

\[\begin{aligned} & f_t = \sum_{i=1}^N \text{PTDF}_{i,b} \cdot \text{Bal}_{i,t}, \quad \forall t \in \{1,\dots, T\} \end{aligned}\]

on which $\text{PTDF}$ is the $N \times B$ system Power Transfer Distribution Factors (PTDF) matrix, and $\text{Bal}_{i,t}$ is the active power bus balance expression (i.e. $\text{Generation}_{i,t} - \text{Demand}_{i,t}$) at bus $i$ at time-step $t$.

StaticBranchUnbounded

Formulation valid for PTDFPowerModel Network model

• FlowActivePowerVariable:
  • Bounds: $(-\infty,\infty)$
  • Symbol: $f$

Objective:

No cost is added to the objective function.

Expressions:

No expressions are used.

Constraints:

For each branch $b \in \{1,\dots, B\}$ (in a system with $N$ buses) the constraints are given by:

\[\begin{aligned} & f_t = \sum_{i=1}^N \text{PTDF}_{i,b} \cdot \text{Bal}_{i,t}, \quad \forall t \in \{1,\dots, T\} \end{aligned}\]

on which $\text{PTDF}$ is the $N \times B$ system Power Transfer Distribution Factors (PTDF) matrix, and $\text{Bal}_{i,t}$ is the active power bus balance expression (i.e. $\text{Generation}_{i,t} - \text{Demand}_{i,t}$) at bus $i$ at time-step $t$.

HVDCTwoTerminalUnbounded

Formulation valid for PTDFPowerModel Network model

This model assumes that it can transfer power from two AC buses without losses and no limits.

Variables:

• FlowActivePowerVariable:
  • Bounds: $\left(-\infty,\infty\right)$
  • Symbol: $f$

Objective:

No cost is added to the objective function.

Expressions:

The variable FlowActivePowerVariable $f$ is added to the nodal balance expression ActivePowerBalance, by adding the flow $f$ in the receiving bus and subtracting it from the sending bus. This is used then to compute the AC flows using the PTDF equation.

Constraints:

No constraints are added.

HVDCTwoTerminalLossless

Formulation valid for PTDFPowerModel Network model

This model assumes that it can transfer power from two AC buses without losses.

Variables:

• FlowActivePowerVariable:
  • Bounds: $\left(-\infty,\infty\right)$
  • Symbol: $f$

Static Parameters

• $R^\text{from,min}$ = PowerSystems.get_active_power_limits_from(branch).min
• $R^\text{from,max}$ = PowerSystems.get_active_power_limits_from(branch).max
• $R^\text{to,min}$ = PowerSystems.get_active_power_limits_to(branch).min
• $R^\text{to,max}$ = PowerSystems.get_active_power_limits_to(branch).max

Objective:

No cost is added to the objective function.

Expressions:

The variable FlowActivePowerVariable $f$ is added to the nodal balance expression ActivePowerBalance, by adding the flow $f$ in the receiving bus and subtracting it from the sending bus. This is used then to compute the AC flows using the PTDF equation.

Constraints:

\[\begin{align*} & R^\text{min} \le f_t \le R^\text{max},\quad \forall t \in \{1,\dots, T\} \\ \end{align*}\]

where:

\[\begin{align*} & R^\text{min} = \begin{cases} \min\left(R^\text{from,min}, R^\text{to,min}\right), & \text{if } R^\text{from,min} \ge 0 \text{ and } R^\text{to,min} \ge 0 \\ \max\left(R^\text{from,min}, R^\text{to,min}\right), & \text{if } R^\text{from,min} \le 0 \text{ and } R^\text{to,min} \le 0 \\ R^\text{from,min},& \text{if } R^\text{from,min} \le 0 \text{ and } R^\text{to,min} \ge 0 \\ R^\text{to,min},& \text{if } R^\text{from,min} \ge 0 \text{ and } R^\text{to,min} \le 0 \end{cases} \end{align*}\]

and

\[\begin{align*} & R^\text{max} = \begin{cases} \min\left(R^\text{from,max}, R^\text{to,max}\right), & \text{if } R^\text{from,max} \ge 0 \text{ and } R^\text{to,max} \ge 0 \\ \max\left(R^\text{from,max}, R^\text{to,max}\right), & \text{if } R^\text{from,max} \le 0 \text{ and } R^\text{to,max} \le 0 \\ R^\text{from,max},& \text{if } R^\text{from,max} \le 0 \text{ and } R^\text{to,max} \ge 0 \\ R^\text{to,max},& \text{if } R^\text{from,max} \ge 0 \text{ and } R^\text{to,max} \le 0 \end{cases} \end{align*}\]

HVDCTwoTerminalDispatch

Formulation valid for PTDFPowerModel Network model

Variables

• FlowActivePowerToFromVariable:

  • Symbol: $f^\text{to-from}$

• FlowActivePowerFromToVariable:

  • Symbol: $f^\text{from-to}$

• HVDCLosses:

  • Symbol: $\ell$

• HVDCFlowDirectionVariable

  • Bounds: $\{0,1\}$
  • Symbol: $u^\text{dir}$

Static Parameters

• $R^\text{from,min}$ = PowerSystems.get_active_power_limits_from(branch).min
• $R^\text{from,max}$ = PowerSystems.get_active_power_limits_from(branch).max
• $R^\text{to,min}$ = PowerSystems.get_active_power_limits_to(branch).min
• $R^\text{to,max}$ = PowerSystems.get_active_power_limits_to(branch).max
• $L_0$ = PowerSystems.get_loss(branch).l0
• $L_1$ = PowerSystems.get_loss(branch).l1

Objective:

No cost is added to the objective function.

Expressions:

Each FlowActivePowerToFromVariable $f^\text{to-from}$ and FlowActivePowerFromToVariable $f^\text{from-to}$ is added to the nodal balance expression ActivePowerBalance, by adding the respective flow in the receiving bus and subtracting it from the sending bus. That is, $f^\text{to-from}$ adds the flow to the from bus, and subtracts the flow from the to bus, while $f^\text{from-to}$ adds the flow to the to bus, and subtracts the flow from the from bus This is used then to compute the AC flows using the PTDF equation.

In addition, the HVDCLosses are subtracted to the from bus in the ActivePowerBalance expression.

Constraints:

\[\begin{align*} & R^\text{from,min} \le f_t^\text{from-to} \le R^\text{from,max}, \forall t \in \{1,\dots, T\} \\ & R^\text{to,min} \le f_t^\text{to-from} \le R^\text{to,max},\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{to-from} - f_t^\text{from-to} \le L_1 \cdot f_t^\text{to-from} - L_0,\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{from-to} - f_t^\text{to-from} \ge L_1 \cdot f_t^\text{from-to} + L_0,\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{from-to} - f_t^\text{to-from} \ge - M^\text{big} (1 - u^\text{dir}_t),\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{to-from} - f_t^\text{from-to} \ge - M^\text{big} u^\text{dir}_t,\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{to-from} - f_t^\text{from-to} \le \ell_t,\quad \forall t \in \{1,\dots, T\} \\ & f_t^\text{from-to} - f_t^\text{to-from} \le \ell_t,\quad \forall t \in \{1,\dots, T\} \end{align*}\]

PhaseAngleControl

Formulation valid for PTDFPowerModel Network model

Variables:

• FlowActivePowerVariable:

  • Bounds: $(-\infty,\infty)$
  • Symbol: $f$

• PhaseShifterAngle:

  • Symbol: $\theta^\text{shift}$

Static Parameters

• $R^\text{max}$ = PowerSystems.get_rating(branch)
• $\Theta^\text{min}$ = PowerSystems.get_phase_angle_limits(branch).min
• $\Theta^\text{max}$ = PowerSystems.get_phase_angle_limits(branch).max
• $X$ = PowerSystems.get_x(branch) (series reactance)

Objective:

No changes to objective function

Expressions:

Adds to the ActivePowerBalance expression the term $-\theta^\text{shift} /X$ to the from bus and $+\theta^\text{shift} /X$ to the to bus, that the PhaseShiftingTransformer is connected.

Constraints:

For each branch $b \in \{1,\dots, B\}$ (in a system with $N$ buses) the constraints are given by:

\[\begin{aligned} & f_t = \sum_{i=1}^N \text{PTDF}_{i,b} \cdot \text{Bal}_{i,t} + \frac{\theta^\text{shift}_t}{X}, \quad \forall t \in \{1,\dots, T\}\\ & -R^\text{max} \le f_t \le R^\text{max},\quad \forall t \in \{1,\dots, T\} \end{aligned}\]

on which $\text{PTDF}$ is the $N \times B$ system Power Transfer Distribution Factors (PTDF) matrix, and $\text{Bal}_{i,t}$ is the active power bus balance expression (i.e. $\text{Generation}_{i,t} - \text{Demand}_{i,t}$) at bus $i$ at time-step $t$.

TwoTerminalLCCLine

Formulation valid for ACPPowerModel Network model

Variables:

• [HVDCRectifierDelayAngleVariable]:

  + Bounds: ``(-\alpha_{r,t}^\text{min},\alpha_{r,t}^\text{max})``
  + Symbol: ``\alpha_{r,t}``

• [HVDCInverterExtinctionAngleVariable]:

  + Bounds: ``(-\gamma_{i,t}^\text{min},\gamma_{i,t}^\text{max})``
  + Symbol: ``\gamma_{i,t}``

• [HVDCRectifierPowerFactorAngleVariable]:

  + Bounds: ``\{0,1\}``
  + Symbol: ``\phi_{r,t}``

• [HVDCInverterPowerFactorAngleVariable]:

  + Bounds: ``\{0,1\}``
  + Symbol: ``\phi_{i,t}``

• [HVDCRectifierOverlapAngleVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``\mu_{r,t}``

• [HVDCInverterOverlapAngleVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``\mu_{i,t}``

• [HVDCRectifierTapSettingVariable]:

  + Bounds: ``(t_{r,t}^\text{min},t_{r,t}^\text{max})``
  + Symbol: ``t_{r,t}``

• [HVDCInverterTapSettingVariable]:

  + Bounds: ``(t_{i,t}^\text{min},t_{i,t}^\text{max})``
  + Symbol: ``t_{i,t}``

• [HVDCRectifierDCVoltageVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``v_{r,t}^\text{dc}``

• [HVDCInverterDCVoltageVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``v_{i,t}^\text{dc}``

• [HVDCRectifierACCurrentVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``I_{r,t}^\text{ac}``

• [HVDCInverterACCurrentVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``I_{i,t}^\text{ac}``

• [DCLineCurrentFlowVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``I^\text{dc}``

• [HVDCActivePowerReceivedFromVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``p_{r,t}^\text{ac}``

• [HVDCActivePowerReceivedToVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``p_{i,t}^\text{ac}``

• [HVDCReactivePowerReceivedFromVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``q_{r,t}^\text{ac}``

• [HVDCReactivePowerReceivedToVariable]:

  + Bounds: [0.0, ]
  + Symbol: ``q_{i,t}^\text{ac}``

Static Parameters

• $R^\text{dc}$ = PowerSystems.get_r(lcc)
• $N_r$ = PowerSystems.get_rectifier_bridges(lcc)
• $N_i$ = PowerSystems.get_inverter_bridges(lcc)
• $X_r$ = PowerSystems.get_rectifier_xc(lcc)
• $X_i$ = PowerSystems.get_inverter_xc(lcc)
• $a_r$ = PowerSystems.get_rectifier_transformer_ratio(lcc)
• $a_i$ = PowerSystems.get_inverter_transformer_ratio(lcc)
• $t_r$ = PowerSystems.get_rectifier_tap_setting(lcc)
• $t_i$ = PowerSystems.get_inverter_tap_setting(lcc)
• $t^\text{min}_r$ = PowerSystems.get_rectifier_tap_setting(lcc).min
• $t^\text{max}_r$ = PowerSystems.get_rectifier_tap_setting(lcc).max
• $t^\text{min}_i$ = PowerSystems.get_inverter_tap_setting(lcc).min
• $t^\text{max}_i$ = PowerSystems.get_inverter_tap_setting(lcc).max

Objective:

No changes to objective function

Expressions:

The variable HVDCActivePowerReceivedFromVariable $p_{r,t}^\text{ac}$ is added to the nodal balance expression ActivePowerBalance as a negative load, since the rectifier takes power from the AC system and to injects it into the DC system. On the other hand, the variable HVDCActivePowerReceivedToVariable $p_{i,t}^\text{ac}$ is added to the nodal balance expression ActivePowerBalance as a positive load, since it takes the power from the DC system and injects it back into the AC system.

The variables HVDCReactivePowerReceivedFromVariable $q_{r,t}^\text{ac}$ and HVDCReactivePowerReceivedToVariable `q_{i,t}^\text{ac}are added to the nodal balance expressionActivePowerBalance as positive loads, since they consume reactive power from the AC system to allow current transfer in converters during commutation.

Constraints:

• Rectifier:

\[\begin{aligned} & v^\text{dc}_{r,t} = \frac{3}{\pi}N_r \left( \sqrt{2}\frac{a_r v^\text{ac}_{r,t}}{t_{r,t}}\\cos{\alpha_{r,t}}-X_r I^\text{dc}_t \right)\\ & \mu_{r,t} = \arccos \left( \cos\alpha_{r,t} - \frac{\sqrt{2} I^\text{dc}_t X_r t_{r,t}}{a_r v^\text{ac}_{r,t}} \right) - \alpha_{r,t}\\ & \phi_{r,t} = \arctan \left( \frac{2\mu_{r,t} + \sin(2\alpha_{r,t}) - \sin(2\mu_{r,t} + 2\alpha_{r,t})}{\cos(2\alpha_{r,t}) - \cos(2\mu_{r,t} + 2\alpha_{r,t})} \right)\\ \end{aligned}\]

Which can be approximated as:

\[\begin{aligned} & \phi_{r,t} = arccos(\frac{1}{2}\cos\alpha_{r,t} + \frac{1}{2}\cos(\alpha_{r,t} + \mu_{r,t})) \end{aligned}\]

\[\begin{aligned} & I^\text{ac}_{r,t} = \sqrt{6} \frac{N_r}{\pi} I^\text{dc}_t\\ & p^\text{ac}_{r,t} = \sqrt{3} I^\text{ac}_{r,t} \frac{a_r v^\text{ac}_{r,t}}{t_{r,t}}\cos{\phi_{r,t}} \\ & q^\text{ac}_{r,t} = \sqrt{3} I^\text{ac}_{r,t} \frac{a_r v^\text{ac}_{r,t}}{t_{r,t}}\sin{\phi_{r,t}} \\ \end{aligned}\]

• Inverter:

\[\begin{aligned} & v^\text{dc}_{i,t} = \frac{3}{\pi}N_i \left( \sqrt{2}\frac{a_i v^\text{ac}_{i,t}}{t_{i,t}}\\cos{\gamma_{i,t}}-X_i I^\text{dc}_t \right)\\ & \mu_{i,t} = \arccos \left( \cos\gamma_{i,t} - \frac{\sqrt{2} I^\text{dc}_t X_i t_{i,t}}{a_i v^\text{ac}_{i,t}} \right) - \gamma_{i,t}\\ & \phi_{i,t} = \arctan \left( \frac{2\mu_{i,t} + \sin(2\gamma_{i,t}) - \sin(2\mu_{i,t} + 2\gamma_{i,t})}{\cos(2\gamma_{i,t}) - \cos(2\mu_{r,t} + 2\gamma_{i,t})} \right)\\ \end{aligned}\]

Which can be approximated as:

\[\begin{aligned} & \phi_{i,t} = arccos(\frac{1}{2}\cos\gamma_{i,t} + \frac{1}{2}\cos(\gamma_{i,t} + \mu_{i,t})) \end{aligned}\]

\[\begin{aligned} & I^\text{ac}_{i,t} = \sqrt{6} \frac{N_i}{\pi} I^\text{dc}_t\\ & p^\text{ac}_{i,t} = \sqrt{3} I^\text{ac}_{i,t} \frac{a_i v^\text{ac}_{i,t}}{t_{i,t}}\cos{\phi_{i,t}} \\ & q^\text{ac}_{i,t} = \sqrt{3} I^\text{ac}_{i,t} \frac{a_i v^\text{ac}_{i,t}}{t_{i,t}}\sin{\phi_{i,t}} \\ \end{aligned}\]

• DC Transmission Line:

\[\begin{aligned} & v^\text{dc}_{i,t} = v^\text{dc}_{r,t} - R_\text{dc}I^\text{dc}_t \end{aligned}\]

Valid configurations

Valid DeviceModels for subtypes of Branch include the following: