Generator

Dispatchable active and reactive power injection with a quadratic cost.

Model Parameters

Symbol	Units	Data	Description	Typical Value	Note
\(P^{\min}\)	[p.u.]	Pmin	Active power lower limit	0	Default unbounded
\(P^{\max}\)	[p.u.]	Pmax	Active power upper limit	1	Default unbounded
\(Q^{\min}\)	[p.u.]	Qmin	Reactive power lower limit	-0.5	Default unbounded
\(Q^{\max}\)	[p.u.]	Qmax	Reactive power upper limit	0.5	Default unbounded
\(c_0\)	[$/h]	c0	Constant cost coefficient		Default 0
\(c_1\)	[$/h/p.u.]	c1	Linear cost coefficient	1000	Default 0
\(c_2\)	[$/h/p.u.²]	c2	Quadratic cost coefficient	100	Default 0
\(u\)	[-]	in_service	Service status		Default true

Parameter Validation

• \(P^{\min} \le P^{\max}\)

• \(Q^{\min} \le Q^{\max}\)

Model Derived Parameters

None.

Model Ports

Name	Port	Description
bus	Bus	Port bus

Model Variables

Internal Variables

Symbol	Units	Description	Note
\(P\)	[p.u.]	Active power into the bus
\(Q\)	[p.u.]	Reactive power into the bus

External Variables

None.

Model Equations

Objective

\[\begin{aligned} f &\mathrel{+}= u (c_0 + c_1 P + c_2 P^2) \end{aligned}\]

Internal Constraints

Equality

Out of service:

\[\begin{split}\begin{aligned} 0 &= P \\ 0 &= Q \end{aligned}\end{split}\]

Inequality

In service:

\[\begin{split}\begin{aligned} P^{\min} &\le P \le P^{\max} \\ Q^{\min} &\le Q \le Q^{\max} \end{aligned}\end{split}\]

External Constraints

Power into the bus, added to its balance:

\[\begin{split}\begin{aligned} \Delta P^\mathrm{bus} &\mathrel{+}= P \\ \Delta Q^\mathrm{bus} &\mathrel{+}= Q \end{aligned}\end{split}\]

Initialization

\[\begin{split}\begin{aligned} P &\leftarrow 0 \\ Q &\leftarrow 0 \end{aligned}\end{split}\]