Bus types: slack, PV and PQ
The key idea
At every bus, four quantities matter: voltage magnitude, voltage angle, active power and reactive power. You fix two of them. The power flow solves the other two. The bus type is simply the name for which two you fixed.
The four quantities
A bus is a node in the network — a point where equipment connects. Every bus has the same four electrical quantities:
| Symbol | Name | In plain words |
|---|---|---|
| |V| | Voltage magnitude | How strong the voltage is at this bus. |
| δ | Voltage angle | How far this bus's voltage wave is shifted in time, relative to the reference bus. |
| P | Active power | The power that does real work — turns motors, heats water, lights rooms. |
| Q | Reactive power | The power that moves back and forth to hold up the voltage. It does no net work. |
A power flow study cannot solve all four at every bus. It needs two of them as inputs. The three standard bus types are the three useful ways to choose those two.
Try it
Select each bus below. Watch which quantities are fixed and which ones the solver finds. Then select the PV bus and move the slider past its limit.
Fixed — you set these
- |V|Voltage magnitude
- δVoltage angle
Solved — the calculation finds these
- PActive power
- QReactive power
The balance point. It supplies whatever power the network still needs — including the losses, which nobody knows until the solution exists. Its angle is the 0° reference.
Slack bus — the balance point
You fix: voltage magnitude (|V|) and voltage angle (δ). The solver finds: active power (P) and reactive power (Q).
Here is the problem the slack bus solves. Power flows through lines and transformers, and some of it is lost as heat. You cannot know the losses before you solve the network — but you cannot solve the network without knowing how much power must be generated. That is a circular problem.
The slack bus breaks the circle. Its power output is left open. After the solver finds all the voltages, whatever power is still missing — the losses, plus any gap between generation and demand — comes from the slack bus. It is the balance point of the whole calculation.
The slack bus also gives the network its reference angle. Voltage angles only have meaning relative to something, so the slack bus is set to δ = 0°, and every other angle is measured against it.
Every electrical island needs exactly one slack bus. No slack: the losses have no source, and the solver cannot balance the equations. Two slacks in one island: the problem has no single answer. In practice, the slack is the grid connection point or the largest generator.
PV bus — the controlled generator
You fix: active power (P) and voltage magnitude (|V|). The solver finds: reactive power (Q) and voltage angle (δ).
A PV bus is a place where a machine actively holds the voltage. You tell it two things: how much power to produce, and what voltage to hold. Power stations, large solar and wind plants with voltage control, and synchronous condensers are all modeled as PV buses.
Holding a voltage is not free. The machine holds it by producing or absorbing reactive power, and the solver finds how much that costs — that is the solved Q.
The Q limit — when a PV bus becomes a PQ bus
A real generator can only produce or absorb a limited amount of reactive power. These limits are called Qmin and Qmax.
Sometimes holding the voltage target would need more reactive power than the machine has. When that happens, the solver does the same thing the real machine would do:
- Q locks at the limit. It is now a fixed quantity.
- The voltage is released. It is now a solved quantity — and it drifts away from the target.
- The bus behaves as a PQ bus for the rest of the solution.
Phasor reports every conversion of this type. The report matters: a generator that has quietly run out of reactive capability is one of the most common reasons a voltage profile fails.
PQ bus — the load
You fix: active power (P) and reactive power (Q). The solver finds: voltage magnitude (|V|) and voltage angle (δ).
A PQ bus is a point where power is simply taken from the network — or injected into it — at a known value, with nothing there to control the voltage. Most buses in a real network are PQ buses: substations, factories, distribution transformers, homes, and small generators without voltage control.
The voltage at a PQ bus is an outcome, not a setting. That is exactly why the power flow exists: to find out what voltage each load actually receives, so you can check it against the allowed band — for example, within ±5% or ±10% of nominal.
Summary
| Bus type | Represents | You fix | The solver finds |
|---|---|---|---|
| Slack | The balance point and angle reference | |V|, δ | P, Q |
| PV | A generator that holds a voltage | P, |V| | Q, δ |
| PQ | A load, or an injection with no voltage control | P, Q | |V|, δ |
Three quick rules that follow:
- One slack per island — always exactly one.
- A PV bus is only a PV bus while its reactive power stays inside its limits.
- The voltages you check against limits are the solved voltages at the PQ buses.
The math, if you want it
Each bus contributes two power-balance equations. At bus i, with voltage magnitudes V and angles δ, and network admittances G + jB:
Pi = Vi Σk Vk (Gik cos δik + Bik sin δik)
Qi = Vi Σk Vk (Gik sin δik − Bik cos δik)
where δik = δi − δk. For a network with n buses there are 2n equations and 2n unknowns — but only if each bus supplies exactly two known quantities. That is what the bus types guarantee. The Newton–Raphson method then adjusts the unknown V and δ values until both equations balance at every bus.
See it in Phasor
In Phasor, you assign the slack, set the PV bus voltage targets and enter the Q limits on the generator elements. When you run a power flow, the violations list shows every bus outside its voltage band — and every PV bus that hit its limit and converted to PQ.